Sugar & Rye is a nice, fairly new restaurant near the Strand. I also recommend Hearsay on the Strand, another new place.
We always hit Galveston Island Brewing (Stewart Road) and the Naked Iguana (Strand) when we're in town.
If you are staying on the west end, Waterman's is a great spot for dinner.
And if you have a sweet tooth, Cordray's Drugstore has great rolled ice cream.
ETA ... the newly remodeled movie theater on the Seawall is pretty nice, especially for a day when the weather is bad. They also have a bowling alley inside.