-Esther's Follies: A vaudeville-like comedy and magic show that is really funny. It's been an Austin mainstay for decades.
-Barton Springs: A natural swimming pool fed by a cold spring. The water is always 67 degrees year-round, making it cold in the summer, and warm during the winter.
-Hamilton's Pool: Another naturally-formed swimming pool, but this one is located inside a grotto (cave that collapsed thousands/million of years ago) and is fed by a waterfall. It's located on a nature preserve, so there's a trail through the woods that you can go through that is also quite nice. Recently, part of the movie Predators was filmed here.
EDIT: try to avoid coming on a weekend. Only 75 cars are allowed to park at the entrance to the park, and others are not allowed in until others leave, which means that you could be stuck in a line of cars for ages in the heat. It's much easier to get in on a weekday. If you can only come on a weekend, try to get there early. Also, bring a cooler with some drinks. There aren't any water fountains or anything at the preserve, and you don't want to get dehydrated.
Alamo Drafthouse: Greatest movie theater in the world. Look through the menu before the show, pick what food and drink (yes, they serve alcohol) you want, write it on a piece of paper, and a waiter will serve it to you during the movie. They also have a lot of awesome events and shows, like Master Pancake Theater, which is like a live version of MST3K.