SEAFOOD:
My favorite is raw oysters at Gaido's at 3828 seawall blvd.
BEACHES:
Water is warm enough to swim pretty much year round. It's different than NE as the slope seaward is very gradual, so the waves churn up more sand, making the water look murky. Once you get out to deeper water, it clears up some.
I never spent much time on the city beaches below the sea wall, but they used to take pretty good care of them.
You can drive your car on most of Texas's beaches. Going west from Galveston are some pretty remote beaches. In 25 miles you get to St Louis Pass, which is where the Galveston Bay empties into the ocean and a state park that's not much.
FERRY:
The ferry from East Galveston to Bolivar Island is free and short. Often there's dolphins surfing the bow wave.
Then U can turn around and come back, OR take US87 east about 25 miles to where the road is closed. From there you can try to drive on the dirt path a ways and enjoy the usually deserted beaches. When your done with that, you can take State Rd 124 to Winnie, TX to Join up with IS-10 back to Houston.
FISHING:
There's public piers that allow fishing on them around town that also sells bait. I never had much luck there as the crabs ate my bait almost instantly.
Also there's several numerous companies that take people fishing for a few hours or all day at mostly modest prices. All have rod/reel rentals at nominal cost, and the crew takes care of anything U catch.