The official city website is pretty comprehensive:
GALVESTON.COM: Official Website of Galveston Island, Texas Tourism & Marketing
Google has a good list for the usual tourist sites:
Some of our favorite eateries include...
The Strand: Star Drug Store, The Original Mexican Cafe, Mosquito Cafe, Olympia Grill
Seawall: The Spot, Salsa's, Gaido's, Mario's Seawall
Just scratching the surface here of the non-chain stuff. The DW and I always stop at the Spot upon arrival; that's always first. Mario's on the Seawall is a little bit upscale Italian, but the best pizza is actually at Mario's Ristorante on 61st; that's the best mom-and-pop Italian joint in town.
This list is pretty damn casual, there are a ton of fancier places to go, but you won't have a problem finding them. We usually stick to the funky, casual stuff.