Thought I would just include a little sidebar about placing a bet. I learned a lot in my one forray to the betting window.
You can pick a horse to either win (come in first) place (come in second) or show (come in third). You're given a book that names the horses running in each race, and if you look at the odds board, you can see the updated odds for the horses in that race.
Note: Place your bets long before the race ever starts. If you wait too long, you'll miss the race for standing in the incredibly long lines. However, I wouldn't place your Derby bet unitl an hour or two before the race, because the odds will change (unless you're like me, and just bet on the horse with the best name. Ha.)
Also, keep your ticket. If you win, you need your ticket. If you don't win, it's a souvenir from your day at the races.