Depends on what types of hotels you like. There are some nice hotels here, but evidently the markup for the F1 period is anywhere from 3-12x the normal rate!
The Downtown part of Austin is pretty small, so if you stay anywhere in Downtown you won't have trouble getting to a shuttle that will take you to the track. The Driskill is a really nice historic hotel and it's right on 6th St, which you'll definitely want to check out after 10PM to go bar-hopping and enjoy the copious amount of live music here covering all sorts of genres. The Omni is a block from there and also very nice, and there's a W and Four Seasons a little farther away. Honestly though unless you're a hotel connoisseur I'd recommend staying at a more basic place near the track and renting a car to get around, or even checking out Craigslist because I know people are renting spare rooms in their houses for that time, or even their entire place if they'll be out of town then. It's a pretty common thing to do out here because of all of the music festivals and such we have (Austin City Limits, South by Southwest, etc) -- but of course use your head if you go that route.
As for what to do, it really depends on what you like. I moved here 3 years ago from SF and love it. Austin is known for its live music (primarily 6th St at night) and restaurants; we eat more meals in restaurants per capita than any other city in the country, apparently. If you want great BBQ, drive out to The Salt Lick in Driftwood and get the brisket and pork ribs. Their vinegar-based potato salad (rather than mayo) and blackberry cobbler are also excellent.
As for general stuff, the UT campus is pretty cool, make sure you're on one of the Downtown bridges around sunset to see the 6 million bats fly out from under it, check out the shops and eat at a food truck in the South Congress area, and see a movie at the Alamo Drafthouse (www.drafthouse.com
) -- they have no ads before films, a full restaurant-grade menu you can order from inside the theater, and even some fun unconventional programming in addition to current releases, definitely a fun and special way to watch a movie.
Austin also has a lot of outdoorsy type activities. There are lots of wineries you can check out in Hill Country, lots of hiking trails and picnic locations (Hamilton Pool), and there's even a whole outdoor adventure park (http://www.hiddenfallsadventurepark.com/atv.html
). Oh, and North Austin has a K1 Speed course (indoor electric go-karts that can reach 45 MPH, pretty sweet) if you get the itch from watching F1 -- or better yet, get a "premium" rental car like a Mustang or Camaro and go enjoy the ENDLESS -- and cop-free! -- twisty back country roads out here.
EDIT: Or if you want to rent an M3 (or at least a BMW), check out this place: http://www.luxurycarrentalaustin.net/
. They're actually a sponsor of the local BMW CCA chapter.