Austin exists because the Republic of Texas wanted a new capital. In 1839 the Texas Congress picked a site on the Colorado River to replace Houston, and the University of Texas held its first classes in 1883. Lawmakers, students and musicians have shared the same bars ever since. The official slogan is The Live Music Capital of the World; the locals' motto, Keep Austin Weird, is now printed on T-shirts.
Three days is the right length. Austin is spread out, and its sights are good but few, none needing more than half a day. What you come for is the rhythm: a breakfast taco, a swim in a spring-fed pool, a long lunch of smoked beef and a night on a dance floor. Three days fits the Capitol area, South Austin and one day trip to Lockhart, the town the Legislature named the barbecue capital of Texas. Add a fourth day only for San Antonio or the wine towns.
Day one is downtown, day two a day trip by car to Lockhart and Wimberley, day three South Austin. More on the country in the United States guide. Start on a Wednesday: Franklin is closed Mondays, Matt's El Rancho Tuesdays, Joe's Bakery Mondays, Barton Springs is shut Thursdays from 9am to 7pm, and the Broken Spoke dances Tuesday to Saturday, so Wednesday puts all of them on an open day.