My parents grew up in Galveston and I’m BOI (Born on the Island) myself. Besides hitting the beach, there are three things that I almost insist on doing when I visit Galveston:
Drive down Broadway and look at the architecture. Ashton Villa and the Bishop’s Palace have tours that are worth the hour or so, but there are also so many private homes that it’s worth stopping and just walking on the sidewalk (especially around 14th Street.)
Hit Benno’s for the pepper shrimp. There’s a lot of fried fish on the menu, but the prices are ok. They have a deck that overlooks the Gulf and is great for sunsets.
Take the ferry to Bolivar peninsula and back. It’s a part of State Highway 87, making it the cheapest two hour cruise anywhere. Go at the end of the day so that you can enjoy the sunset on the way over (and maybe see a dolphin or two.) Drive by the Bolivar lighthouse if you want, and don’t forget to bring crackers to feed the gulls from the stern, or stand at the bow and let the waves crash over you.
For fine dining, I’ll second (third? fifth?) Gaido’s. Family story: Gaido’s used to be closed on Mondays (they’re open seven days a week now.) Back in the 1940s, an aunt of mine made a running joke of asking her brother when he would take her to Gaido’s. He would reply, “Oh, how about Monday?” This went on for five years or more until a hurrican blew through over a weekend. To make up revenue, Gaido’s stayed open the next Monday, and Aunt Evelyn got her dinner.
Go to the Strand, but don’t spend a lot of time in the shops. Instead, hit:
The Galveston Historical Foundation. The last time that I checked, they had a good exhibit about life in Galveston when it was the major seaport for Texas. Of course, they also have exhibits and a movie about the 1900 Storm (You may also want to check out www.1900storm.com.)
I volunteered on Elsissa when she had just been restored. She’s worth checking out. There’s also a small museum there, and the pier doubles as a not-quite historical reenactment site since most maintenance is done there (though they probably cut back on the weekend work these days.)
Colonel Bubbie’s is a pretty cool place to rummage through, though I haven’t found anything that I needed there. I understand that the owner made a killing selling uniforms for Saving Private Ryan.
There is also a flight museum at Scholes Field. I’ve never been there, but it’s associated with the Commemorative Air Force would probably be worth the visit.
Any beach is a good one, but I like the pocket parks west of town and far east Stewart Beach when it isn’t too crowded. You can also drive down the beach in Brazoria County and they’re almost empty at night, but that’s the better half of an hour away over San Luis Pass.