First, you need to decide whether you are a true race fan or did you just come along with someone else and wanted to see if Indianapolis is the cultural desert it is perceived to be by east and left coasters.
Assuming you are the former, there is a race going on somewhere every night. Try the Speedrome on South Kitley for the Figure 8 race. But, be very careful to not touch anything and make sure you are up to date on all of your vaccinations. And I am sure there will be some kind of sprint car race at Raceway Park. It is somehat safe to eat at the concession stand if your are taking Lipitor. Most importantly, go to the Speedway Museum at the IMS. It has an unbelieveable collection of not only racing cars from the 500, but contains some of rarest and most valuable classic cars of all time.
If you are not a racing fan, Indy can provide all of the cultural events you would expect from a large city. Hey, it's not New York, but when the Lilly Endowment drops millions every year to support the arts, it can be a very enjoyable to visit the IMA, IRT, IHC, or the IAG. Just in case you didn't notice everything here starts with an "I" because the only two famous people from Indiana were Abraham Lincoln and Dan Quayle. Lincoln just barely passed through on his way to Illinois and Quayle just barely passed English on his way to graduate from Depauw.
Anyway, come Saturday night, you will find yourself in a steak house clogging your arteries while smoking a cigar, at a cookout eating BBQ, or in the Coke lot next to Georgetown Road drinking whatever the guy parked next to you has in his cooler. Everyone shares because they all brought more than they could possibly drink and are tired of carrying it around. I've never spent the night in the Coke lot, but I have seen the beginning of "Slumdog Millionaire". I think they shot some of the footage at the Coke lot.
As for Race Day, dress in layers. The temperature can feel like it changes 40 degrees from the morning until the checkered flag. Plus, there is always a threat of rain so pack a 55 gallon trash bag. When it rains at the Speedway all humanity ceases to exists. You won't care what you look like. You just want to stay dry and warm. Believe me. The trash bag works.
You also need to plan your food for race day. It's a long day, so eat a good breakfast and pack three sanwiches, two candy bars, your favorite chip snacks, cheese sticks, six chocolate chip cookies, four beverages, handy wipes, 30 SPF sunscreen, and wet paper towels to clean up the bird droppings off of your seat. This list has been perfected over the past five decades. It works!
One last thing. Do not cheer for Helio Catroneves. He should be in prison for cheating on his taxes.
Enjoy the race!