Surfside and Surfside Beach, Bryan Beach, and Quintana Beach are south of Lake Jackson on TX-36. There are wide stretches of beaches with vehicle access and few people (at least compared to Galveston). The fishing is often very good, especially for sharks and redfish.
West Columbia has some interesting buildings about a hundred years old.
Palacios is about a half hour south on TX-35 and has lots of old interesting houses and interesting restaurants.
Galveston makes a great day trip and scenic drive. Head north on TX-35 to Lake Jackson, then south on TX-36 until you come to the Bluewater Highway (CR-332). Head north east (left) and follow the coast. The toll bridge at San Luis Pass costs about $2. The road then becomes Termini (CR-3005) and continues to Galveston.
Houston is a major city with endless things to see and do, and is about two hours north on TX-35.