Pretty much anywhere.....Angleton is a fishing paradise.
For freshwater, most of the best is privately owned and requires membership, but the Brazos and San Bernard Rivers can be good, although muddy right now.
Saltwater, just about everywhere. Closest is Bastrop Bayou. Better have a GPS to find your way around as its convoluted. Around the reefs, flats, and shorelines. If you see a flock of gulls working close to the water, go there and throw under them, a fish every time.
Chocolate Bayou is a bit North of Angleton, towards Alvin, it is also a great place to fish the bayous and bays.
At the ocean is Surfside Beach and the Surfside jetty, both great places to fish without a boat. If its too windy and there are a lot of waves, its a good place to body surf.
Within your hour range is San Luis Pass, from there you can fish a lot of West Galveston Bay along the North/West shoreline of the barrier island.
The wind can be relentless, but there are places you can get to and fish even when its windy.