That is, apart from the town's Museum of the Gulf Coast. Covering the area between Houston and New Orleans, it's as fine a regional museum as you can find. The displays explaining the area - and its history as a pirate-infested "no man's land" owned by the French and a buffer between Spanish Texas and the eastern US states - are immaculately curated, but most foreign visitors come for the Music Hall of Fame.
The room illustrates the region's musical heritage by focusing on its role as the cradle of zydeco and swamp pop and boasts significant local movers in country (George Jones) and blues (Clarence "Gatemouth" Brown). But, as with most rock'n'roll tourist sites, the big draws are dead musicians, in this case JD Richardson and Janis Joplin. The former, better known as the Big Bopper, had a hit with "Chantilly Lace" and is commemorated by a huge bronze bust and a cabinet of personal effects - dice, a bottle of aspirin, a Zippo lighter and a hairbrush - recovered from the 1959 plane crash in Iowa that also killed Buddy Holly and Richie Valens.
Even more space is devoted to the hometown girl Joplin, who was born (19 January 1943) within strolling distance of the museum. A replica of her favourite Porsche is parked in a corner overlooked by several of her paintings, and a large cabinet is filled with gold records, stage costumes and other memorabilia. An entry in her school yearbook noting that she was a member of the Slide Rule Club didn't fit with the tales of her rock'n'roll libido and the drug penchant that killed her in an LA motel room aged just 27.
It was a grim thought on which to leave the city, and the overcast weather added to the chill. Typically, I got a bargain upgrade to a Chrysler Sebring convertible on a day when the weather was terrible. The plan was to have the roof down, but what should have been a scenic drive along the Bolivar Peninsula was more a test of windscreen wipers.
In some respects it was a fitting way to enter Galveston though. Few tunes are as evocative as that one named after that town, one of a trilogy of great songs composed by Jimmy Webb and sung by Glen Campbell. As the song goes, the city had its "sea winds blowing".
Occupying the northern part of Galveston Island - a 32-mile barrier reef never much more than two miles wide - the city is known as the "Queen of the Gulf" and the "Wall Street of the south west", as well as being the second-busiest immigration station after New York's Ellis Island. The city boomed until 8 September 1900, when the nation's largest natural catastrophe occurred, and a storm ripped through the city destroying a third of all buildings and killing 6,000 of its 38,000 inhabitants.
Galveston's role as the financial powerhouse of the region was taken up by Houston, but today it has pulled itself together rather well. Sprawling petrochemical plants pump plumes of dodgy-looking smoke into the air around the harbour but its southern side is taken up with beaches and a good number of 19th-century homes and buildings that are open to the public. The old warehouses are now occupied by outlet stores, surprisingly good souvenir shops, craft merchants, art galleries and grand hotels.
With the smattering of small museums and the Moody Gardens overlooking pretty Offat's Bayou, there's plenty to do in and around downtown. But it's mostly a weekend destination. During the week the bars are pretty empty, though the 75-cent drink specials give you a lot of bourbon and coke for the price of a pint back home.
At night you'll find another shrine to a dead Texan musician. The cosy little Old Quarter Acoustic Cafe pays homage to the songwriter Townes Van Zandt, who died in January 1997. He may not be a household name outside east Texas, but Van Zandt is revered by his contemporaries in the folk and country worlds.
You'll find more pictures of him on the wall than drinkers in the bar, and most visitors come for the "open mic" night. It's an indication of how important music is here, and a reminder that Galveston is a city that deserves to have a fine song written about it.
The writer is co-author of Fodor's `Rock and Roll Traveler USA' (pounds 12.99)