@REI-JohnJ has some good suggestions, but right now all of those are very busy and can get extremely crowded. I suggest looking into the state parks a little ways outside of Austin - Inks Lake State Park has some great trails and options for swimming after. McKinney Falls State Park and Lockhart State park are both good hiking options with a variety of levels of difficulty.

The great thing about the state parks is they are limiting the capacity right now, so unlike the local or county parks, you don't run the risk of them being overcrowded. You do, however, have to go to the Texas Parks and Wildlife website to reserve a day pass (usually $6 or less).