I lived down in Austin for three years before moving in May and have played most of the courses in the area.
Assuming you don't have access to Austin CC or Austin GC, Barton Creek is the place to play. To my knowledge, you don't have to stay there to play, though they do charge you an arm and a leg as a non-resort guest. If you don't mind forking over the cash (probably $200-$250), I'd play the on-site courses in this order: 1) Fazio Foothills, 2) Fazio Canyons, 3) Coore and Crenshaw.
Falconhead is a fun, challenging and well-conditioned layout that should run you about $60. Avery Ranch is a step down from Falconhead but pretty decent.
The Bandit and Delaware Springs are good values but IMO don't merit flying down your clubs and then making the hour drive from Austin. Some would disagree.
Lions on Lake Austin Blvd. is a classic muni with wonderful views of the Hill Country, though long rounds are the norm there.
Have a great time down in Austin. That shouldn't be hard, especially with ACL going on. Ah, I'm getting jealous just thinking about it.