There's also a Red's South off 71 toward Oak hill. If you want to do skeet shooting there's capital city clays. Other than that you have to get a goof 40 minutes out of austin for anything else, AFAIK.
What part of town? North there is Eagle Peak that is nice, but you have to "join" which is an extra $10 the first visit. Red's is in P-Ville and at the Y in Oak Hill. Austin Rifle Club is out near Manor, but you have to know a member to shoot there. Hey, I'm a member! This is (obviously) my favorite place. There is also Stan's north of Liberty Hill. Those are all of the ones that I have been to, but there are a couple more as well.
I used to go to Dietz between New Braunfels and Gruene. It's a good range, but a bit of a drive. Nice folks, standard range rules. They do give discounts for NRA members.
I now go to Lone Star Gun Range in Lockhart and have a membership there. Matt is continuously making upgrades, and currently has two pistol bays and a 100y rifle range. Last visit, he'd made a shotgunning range for shooting clays. Controlled rapid fire is allowed, and if the right pistol bay is free, you can draw, move, and shoot there. All of their rifle and pistol positions are covered, as well. I like it a lot.
I don't particularly like indoor ranges, and both listed above are outdoor. I've been to both Red's locations. The one in Pflugerville is nicer, IMO, but still suffers with the same issues most indoor ranges do - lots of noise, either too hot or too cold, and expensive range fees ($13/hour). The outdoor ranges above have about the same price for outdoor shooting that you'll pay for an hour of shooting at Red's.
People have mentioned Eagle Peak above. Their fees are reasonable, but I do not go there because I cannot afford to shoot rifles there. They restrict rifle ammo to SP or HP ammo only, and I cannot afford to buy and feed my milsurp rifles with the kind of ammo they require. Their policies on shooting and handling are also relatively restrictive, like most other public ranges. They do have a shotgun setup, but my understanding is that it is rudimentary at best, and clay throwers are manually operated.