Aquarium at Moody Gardens is great.
Shrimp PoBoys at ShrimpNStuff and eat them down at end of East Beach watching the tankers come in and walk on the WW2 gun turrets.
Grab a loaf of bread and feed the seagulls.
Take the ferry to Bolivar and go 14 miles for BBQ crabs at Stingerays or 22 miles to the nude beach.
Hang out on The Strande with bars and shops, etc.