Brazosport is the collective name for a cluster of small cities south of Houston in Brazoria County, including Lake Jackson, Clute, Angleton, Freeport, and the beach communities of Surfside and Quintana. If you want Gulf Coast beach time without the crowds and commercial sprawl of Galveston, this is where to go. Surfside Beach is the closest ocean beach to Houston proper, and on a weekday you can have long stretches of sand mostly to yourself.
Quintana Beach County Park is the standout, a well-maintained park with a fishing pier, nature trails, and calm, shallow water good for families. The Brazoria National Wildlife Refuge south of Angleton is excellent for birdwatching, especially during winter migration when snow geese arrive by the hundreds of thousands. Sea Center Texas in Lake Jackson is a free aquarium and the world's largest redfish hatchery. The Brazosport Center for the Arts and Sciences in Clute has a small planetarium and natural history exhibits. Surfside Beach is four miles of open Gulf shoreline popular for surf fishing and beach days.
Freeport harbor has a working shrimping fleet, and you can buy fresh shrimp and fish dockside or eat at one of the no-frills seafood restaurants along the harbor. The area is not polished or touristy, which is part of its appeal. It is a real working coast with genuine Texas beach culture: fishing, crabbing, and sitting in a lawn chair watching the pelicans.