The Speedway, home of the Indianapolis 500 motor race, is Indy's supersight. The Hall of Fame Museum features 75 racing cars (including former winners), a 500lb Tiffany trophy and a track tour ($8 extra). OK, so you're on a bus for the latter and not even beginning to burn rubber at 37mph, but it's still fun to pretend.
The big race itself is held on the Sunday of Memorial Day weekend (late May) and attended by 450,000 crazed fans. Tickets can be hard to come by. Try the pre-race trials and practices for easier access and cheaper prices. The track is about 6 miles northwest of downtown.