Hi LEF, couple of things. First, they used to have a couple of nice piers in Galveston that are now gone, wiped out by the hurricane. They're supposed to start rebuilding them soon - not sure if they'll be done by this winter. I was just down there a few weeks ago and the town is just now regaining the look it had before. There are still quite a few shops that haven't reopened, especially the ones in downtown and closest to the water - those were the places hardest hit.
Most people in Galveston now fish the jetties. I'm sure most people have favorites but I find them to all be about the same. My favorite place to surf fish down there is actually San Luis pass. It's about a 20 minute drive from downtown Galveston. There are bait shops in Galveston and there's also one on the highway on the way out to San Luis Pass. It's called The Rusty Hook. The family that runs it lives onsite. They're open from 5am to midnight during the summer. Not sure about winter hours. Once you pull off the highway you drive straight onto the beach - people do it with cars but I personally wouldn't do it. You can fish on the bay side or the gulf side. Gulf side you can expect bull reds, sharks, and the occasional ling. Hope all that helps.
Also, I don't know if this kosher or not but you might google Texas Fishing Forum. They have forums like these where the guys might be able to give you more specific information if you're looking for it.