Lived in Austin 10 years. I can tell you there are lots of great places to live. I will start with the best, and oh some are rather pricey (I actually bought there).
Hyde Park (Just North of UT Campus between 29th and 51st street and south of lamar) The homes can go for 300K to over 1 Mil. Though pricey, you cannot get a more hometown feel. Very nice, lots of bikers, runners, older college students, and perpetual students in the area. Lots of churches, festivals, and the proximity to everything is top knotch.
tarrytown
Just west of westcampus UT inbetween lamar and mopac. I really consider it to carry over mopac to exposition. The homes go for 300 to well over 5 million. One of the most expensive parts of town, but if you drive it you will understand why. Awesome proximity to everything (not that austin is big) and great trees, winding roads, and the like. It really sits on top of a hill, so you get some great views of the hill country.
south austin (Travis Heights) This is not that far south, but across the water from downtown on the west side of riverside dr. It runs from I-35 to Zilker park and as south as oltorf. The prices are 180K to about 600K. I really like this area especially as it got closer to Barton Springs road, by the restaurants. It starts in a valley, and goes on to an area where the houses are on top of hills. If you are allergic to mountain cedar (yes mountain cedar) don't go near this area. It is like Mecca for mountain cedar 3 months a year (part of Feb/mar and again in late sept/oct). Here you get the view of the water as the river is very close, and you are still very close (3-5 miles from downtown).
Jester Heights off 360 This is in the hills back off of 360. You are getting further out of town, but the homes quality increases. It is pricey don't get me wrong, and the average home is probably in the 330's, but some go for as low as 180. Vista views of Austin are out of everyones backyard here.
Westlake Texas The older money, homes in that city go for 150- $$$$Millions. It is very close to downtown (less than 8 miles) and has one of the best high schools in the State. In Texas Austin Westlake is known.
Oh and for the idiot who is talking aobut the republican-filled state, This is austin. If you look at the demographics, you will see it voted Democrat. There are 150,000 college students in that city. From Texas State, Univeristy of Texas, Houston Tillitson, Concordia, St. Edwards University, and Austin Community College system. This is not your regular Texas, that is why everyone makes such a big deal about the city. Please get your facts straight before coming off the cuff.