Roger,
To add to SW's comments, I happen to live in the downtown area. There will be a large variation in what you will spend depending on what area of town you stay in. SW mentioned some of the pricey places to stay, and the location / quality can't be beat!

And he gave great restaurant recommendations as well.
If you want to stay a little further out from downtown to save some money, the track is past the airport on the southeast side of downtown. There are hotels closer to the airport which may be a bit cheaper.
I haven't attended an event at the track yet, so I can't speak to the specifics of getting in/out of the event. I know that the F1 event had a pretty good system running of shuttles/buses which took people back and forth. I don't know if they will do that for any of the other events, but I doubt that they will (they didn't for the recent Sports Car race there).
As for bars, the most famous 'district' of Austin is 6th Street. However, I would only recommend that you take a walk down it and see if it's your thing or not. The crowd can be very diverse, and it's more of the college-age area. There will be a lot of live music options at all of the bars along 6th, and those can be a lot of fun, again depending on your tastes in music. We have a laid-back new area called Rainey St. which has some great outdoor bars. You'll find music there also, along with some great 'trailer food'. 4th and 5th Streets also have some good bars and restaurants (a couple which SW mentioned), and the west side of 6th Street as well.
All of the above is within easy walking distance of the downtown hotels.
South Congress Avenue is a fun place to walk at night. A popular destination there is the Continental Club, a local bar whose owner is a hot rodder. You will often find a nice array of cars parked along the street outside of the club. There are some good restaurants along that strip as well (South Congress Cafe, Vespaio, Botticelli's, Guero's). Also, a cool hotel to stay right in that strip is Hotel San Jose. All of this is a relatively short walk to the areas above also, or a $7 cab ride.
Hope all of this helps! Austin is a fun place, I hope you enjoy the visit!