The American map does not always make it easy for those who want to learn more about the Western Hemisphere's most eulogised country.

Take its capital cities as an unwieldy example. The big one - the one with all the monuments and grand buildings; Washington DC - sits entirely outside the state system in its own District of Columbia.

And many of the 50 individual states beyond it like to flatter to deceive. They will show you, on one hand, a dazzling metropolis - a cosmopolitan giant that, you assume, is the king of the region around it - while, at the same time, they are squirrelling away their administrative machinations in a quieter corner of their territory.

It is like an end-of-the-pier sleight-of-hand conjuring trick - only one involving skyscrapers and soaring bridges rather than rabbits from hats.

Look at New York. It is the most famous city on the planet. A labyrinth of music, food, culture, history and architecture. It is the most obvious case of an American city sharing its name with the wider state in which it sits. But is it the capital of said state? Ah no, sir. Madam, I am afraid not. Not even close.

True, there are several incidents of a state's biggest city also being its official centrepoint. Boston has long been the urban star of Massachusetts, Denver gets the gig in Colorado, Atlanta does the honours in Georgia, Salt Lake City has the job in Utah, Nashville walks the line in Tennessee, Honolulu is top dog in Hawaii and Indianapolis (which almost joins New York on the same-name-as-its-state pantheon) puffs on a cigar as the big boss in Indiana.

But elsewhere, if it looks large enough to be the state capital, if it sounds like a state capital, and if it swaggers like a state capital... then it probably isn't.

Rarely do these apparent incongruities amount to an accident of history. In some cases, they are part of an ongoing attempt to ensure that power is spread across a region, and not just confined to the sparkling urban hubs of finance and finesse where the Instagram eye falls.

In others, capital status is evidence of waning economic fortunes; a high tide mark for a place which was master of all it surveyed amid the frontier frenzy or goldfield gleam of the 19th century, but has been left behind subsequently (in terms of size and fame at least) by some upstart which found its growth spurt at a later date.

Still, this is all grist to the travel mill, providing a reason to glimpse a series of cities you had never heard of, let alone considered visiting.

Or is it? Is it really worth spending some of your hard-earned holiday time exploring a city which has the temerity to be the capital of New York while not being the Big Apple? The following feature has (some of) the answers.

California

The world thinks the capital is: Los Angeles or San Francisco

It actually is: Sacramento

Worth visiting? Absolutely. Though smaller than LA and San Francisco (and indeed, than its state colleagues San Jose and San Diego), Sacramento is hardly a little fish in a big pool. Home to over half a million souls, it swells out around the capitol building - a gorgeous confection in granite, completed in 1874, that could almost be St Paul's Cathedral.

Its History Museum (sachistorymuseum.org) looks back across the best part of two centuries - not least to the California Gold Rush of 1848-1855, the spark for the city's birth. But it revels in the present in the boutique shops of its central "grid" (sacramento.downtowngrid.com), and in the lively bars of J and K Streets. And it is easy to reach.

As with much of California, good highways mean nothing is too far from anywhere. Sacramento sits 90 miles and two hours' drive north-east of San Francisco.

How to do it: America As You Like It (020 8742 8299; americaasyoulikeit.com) sells a two-week "Northern California Highlights" road trip that spends two nights in Sacramento. From £1,466 per person, including flights, accommodation and car hire.

Further information: visitsacramento.com; visitcalifornia.com

New York

The world thinks the capital is: New York City

It actually is: Albany

Worth visiting? Definitely. As with the Big Apple, the history of Albany begins - at least in the colonial era - with Dutch settlers, who founded the fur-trading outpost of Fort Nassau on the banks of the River Hudson in 1614. But there the similarities end.

The New York capital is home to fewer than 100,000 people (as opposed to the eight million crammed into the five boroughs of New York City), and has retained a pretty character as a result. Washington Park is one of many green spaces which shape the centre - it even hosts a Tulip Festival (albanyevents.org) every May that pays tribute to the city's Dutch heritage.

The Albany Institute of History and Art (albanyinstitute.org), meanwhile, is a treasure trove of American painters. It places particular emphasis on the "Hudson River School" of artists (the likes of Thomas Cole and Frederic Church) whose landscape brush-strokes captured the surrounding area in the mid-19th century.

How to do it: Albany is just 150 miles north of New York, and accessible from it - especially by water. American Cruise Lines (001 800 460 4518; americancruiselines.com) offers voyages along the Hudson from the Big Apple - including a series of eight-day ventures (in September and October) that focus on autumn foliage along the river banks. From $4,210 (£3,187) per person (flights extra).

Further information: albany.com; iloveny.com

Florida

The world thinks the capital is: Miami

It actually is: Tallahassee

Worth visiting? In a sense. Florida is eternally defined as a place by the sea - the sun-swathed peninsula with shoreline on both the Atlantic and the Gulf of Mexico. But for all this, its capital is found nowhere near the ocean.

Instead, Tallahassee lurks 40 miles inland, high up in the "Panhandle" where the state begins its journey west towards Alabama and the humidity of the Deep South.

The city has been the Florida kingpin since 1824, and wears its status in what is surely America's most architecturally mis-matched capitol building. The Old Capitol, built in 1845, is a traditional piece of domed classic revivalism - but it is now rather dwarfed by the (frankly ugly) tower that replaced it in 1977.

The first structure is preserved as the Florida Historic Capitol Museum (flhistoriccapitol.gov), and offers an engaging picture of the last 200 years. The cafes, galleries and shops of Railroad Square Art District (railroadsquare.com) might also detain you in an out-of-the-way city that few European travellers consider. 

How to do it: Realistically, you might visit Tallahassee as part of a stay on the top edge of Florida's Gulf Coast. A seven-night break at the Wyndham Garden hotel at Fort Walton Beach (150 miles west of Tallahassee), leaving Heathrow on May 18, starts at £1,197 a head with My America Holiday (0208 290 9751; myamericaholiday.co.uk).

Further information: visittallahassee.com; visitflorida.com

Nevada

The world thinks the capital is: Las Vegas

It actually is: Carson City

Worth visiting? More than you might think. Pinned almost to the outer edge of Nevada, not so far from the California state line - but definitely quite far from Las Vegas, which is almost 500 miles away to the south-east - Carson City is an engaging relic of the American West.

Kit Carson was a frontiersman, and one of the first non-indigenous Americans to visit (what is now) the Eagle Valley, when he arrived in the area in 1843. The town boomed quickly following the discovery of gold and silver in 1859, and was a presence on the map long before Vegas (which was founded in 1905).

The Nevada State Railroad Museum and the Nevada State Museum (see nvculture.org/museums in both cases) preserve various chunky slabs of 19th century life (the latter is set up in the former Carson City Mint) - and there is more than enough to justify a day or so in town en route to the casinos (and superb ski resorts) of Lake Tahoe, a mere 20 miles away.

How to do it: Bon Voyage (0800 316 3012; bon-voyage.co.uk) offers "Neon, Nature and National Parks - a 14-day westward road trip through Nevada, Utah and California which stops off at Carson City. From £2,159 per person - including flights and car hire. 

Further information: visitcarsoncity.com; travelnevada.com

Louisiana

The world thinks the capital is: New Orleans

It actually is: Baton Rouge

Worth visiting? Without doubt. New Orleans may steal the plaudits and much of the international visitor's attention when it comes to Louisiana, but the state's second city can certainly compete when it comes to food, nightlife and music (the city is one of the heartlands of Cajun culture) - and without the great numbers of bead-clad evening revellers who seem to pack out the latter's giddy French Quarter in most corners of the calendar.

The Louisiana Art & Science Museum (lasm.org) has a fine collection of contemporary works by 20th century artists from across the state (such as Conrad Albrizio and Fritz Bultman), while the Red Stick Farmers Market (breada.org) delights in local produce.

You can find busy bars on Government Street where it runs west towards the River Mississippi - and, like New Orleans, Baton Rouge throws itself into Mardi Gras celebrations with gusto. Of course, if you feel it doesn't match up to the Big Easy, you can always flit to the real thing, just 80 miles away to the south-east.

How to do it: Complete North America (0115 961 0590; completenorthamerica.com) sells "New Orleans and Cajun Country" - an exhaustive 14-night tour of Louisiana which stays two nights in Baton Rouge. From £2,899 per person - with flights and car.

Further information: visitbatonrouge.com; louisianatravel.com

Washington

The world thinks the capital is: Seattle

It actually is: Olympia

Worth visiting? You can barely miss it. Olympia sits just 60 miles south-west of Seattle, and if you are taking on the full circuit of Washington's most famous road trip - the loop around the mountains of Olympic National Park - you are all but guaranteed to pass through it.

There is enough to detain you if you do. Home to a mere 50,000 residents, Olympia basks in that slightly left-field arty glow that bathes much of the west coast of the USA. Its Farmers Market (farmers-market.org) is a hive of organic fare, Last Word Books (lastwordbooks.org) is a temple to literature in second-hand form, and Madsen Family Cellars (madsenfamilycellars.com) pours wines produced in the region. This is small-town America as much as a city - and all the more alluring for it.

How to do it: Olympia is easily seen via the 14-day "Northwest Explorer" road trip from Seattle to San Francisco, as sold by the American Road Trip Company (01244 342 099; theamericanroadtripcompany.co.uk). From £1,799 per person, including flights and car.

Further information: experienceolympia.com; experiencewa.com

Oregon

The world thinks the capital is: Portland

It actually is: Salem

Worth visiting? Why not? For Seattle and Olympia in Washington, read Portland and Salem in next-door Oregon. It's much the same principle - an arty big city which dominates perceptions of the state; a smaller capital where life flows around the rituals of local government amid coffee shops and galleries.

And you can barely see one without tripping over the other. Salem lies just 52 miles south of its more eulogised neighbour - and sits on the same river, the Willamette. Said waterway gives it a pretty context - one where the Willamette Queen paddle-steamer splashes away from the grassy shore of Riverfront City Park.

You can wander amid echoes of the 19th century in the central Historic District, investigate the city's roots at Willamette Heritage Center (willametteheritage.org) and wander along leafy trails in Bush's Pasture Park - and still be back in Portland in time for dinner, a craft beer and a hipster comedy show.

How to do it: The 14-night "Washington State and Oregon" road trip sold by Bon Voyage would be an opportunity to pause in Salem. From £2,105 a head, with flights.

Further information: travelsalem.com; traveloregon.com

Pennsylvania

The world thinks the capital is: Philadelphia

It actually is: Harrisburg

Worth visiting? That depends entirely on how intrigued you are by Pennsylvania as a destination. A road trip between the state's two most fabled cities - Philadelphia (a key spot in the genesis of the USA, where the Liberty Bell stands as a symbol of freedom) and Pittsburgh (the birthplace of Andy Warhol; home to a magnificent museum in his honour) - would almost definitely take you through it.

Harrisburg cannot match either of these rivals for history or art, but it is a confident, thriving place on the bank of the Susquehanna River.

The Pennsylvania Capitol is a striking affair - a Beaux Arts delight completed in 1906. City Island is a quirky delight - an outcrop in the middle of the river with a miniature golf course (watergolfcityisland.com). And there are eateries aplenty along the "Restaurant Row" of Second Street. All very enjoyable.

How to do it: A basic seven-day road-trip package for two people - flying from Heathrow to Philadelphia on May 11, and back from Pittsburgh on May 18 - costs from £622 a head, including car hire, via British Airways (0344 493 0787; ba.com). 

Further information: visithersheyharrisburg.org; visitpa.com

Michigan

The world thinks the capital is: Detroit

It actually is: Lansing

Worth visiting? It is only in the last five or so years that international tourists have reconsidered Detroit as an option, so it should perhaps be no surprise that Lansing does not scream its name from the map of Michigan.

But there it is, roughly at the heart of the boxing-glove-shaped fist of land that shapes the lower portion of the state - a kernel of government offices and day-to-day duties which cares little that the world does not know its name.

Still, if you are heading north towards the parts of Michigan that road-trippers do know about - the Straits of Mackinac, where two of the Great Lakes (Huron and Michigan) meet; Mackinac Island, snoozing in the sun nearby - the road may well take you through Lansing.

And if it does, you can admire a grand 19th century capitol building which looks, at first glance, more like a church than an administrative hub, a transport museum that - inevitably in Michigan - gazes back at the early years of the motor car (reoldsmuseum.org), and a modern art museum (broadmuseum.msu.edu) of impressive pep.

How to do it: Audley Travel (01993 838 755; audleytravel.com) sells a 16-day "Classic Great Lakes Self-Drive" around Lake Michigan which can be tailored to cool its tyres in Lansing. From £2,780 per person, including flights (in and out of Chicago).

Further information: lansing.org; michigan.org

Texas

The world thinks the capital is: Houston, Dallas or San Antonio

It actually is: Austin

Worth visiting? Oh yes. With a caveat. For it would be untrue to say that the planet doesn't know that Austin is the Texan kingpin. Indeed, it is definitely the most famous regional capital listed in this feature.

But when the state of which you are head boy is home to three of the biggest cities in America, and you are not one of them (Austin is actually the 11th biggest US city - Houston (fourth), San Antonio (seventh) and Dallas (ninth) all make the top ten), it would be easy to believe you are being overlooked.

Not that Austin is short of self-belief. Or visitors. It markets itself as the "Live Music Capital of the World". It backs up this boast in venues like the Paramount Theatre (austintheatre.org) and the Cactus Cafe (cactuscafe.org) - and in events such as its yearly festival extravaganza South By Southwest (sxsw.com).

Beyond this, there is food (Downtown is full of restaurants such as Texas Chili Parlour; txchiliparlor.com), art (the modern creativity of The Contemporary Austin; thecontemporaryaustin.org) and literature (the Texas Book Festival is held every October; texasbookfestival.org).

The Texas Capitol, meanwhile, is as grand and glorious as you would expect of the Lone Star State - a Renaissance Revival masterpiece, finished in 1885, that stands tall.

How to do it: Hayes & Jarvis (01293 832 699; hayesandjarvis.co.uk) sells "Texas Time" - a nine-day itinerary which ticks off Dallas, San Antonio and Austin in one getaway. From £2,189 per person, including flights, accommodation and car hire.

Further information: austintexas.org; traveltexas.com