The easiest way is to park at Papa John's Stadium for $15 and then either walk to Churchill Downs (15 minutes or so) or catch one of the shuttles they have running back and forth (another $15/person). It's an easy walk but I highly recommend wearing your comfy shoes and then slipping on your heels when you reach Churchill. The hardest part of the walk is the rather lengthy staircase you have to climb from the parking lot to Central Ave but then it's literally downhill from there. You'll pass people selling flip flops, beer or beads, at least two guys with bullhorns talking about Jesus (and at least two wiseguys heckling them), a carnival stand selling everything from funnel cakes to corn dogs, etc.
The first gate you'll come across is Gate 3 on 4th Street. This is where you want to enter if you are sitting in the infield and have a chair, blanket or tarp. The entrance will take you right to the infield. If you have grandstand tickets, keep walking down Central Ave and enter through Gate 1. Blankets and tarps are allowed but not chairs. If you have reserved clubhouse seats, Gate 10 is apparently the place you want to go. No chairs, blankets or tarps are allowing through Gate 10.
Churchill Downs is selling parking passes too. Passes are good for both Oaks and Derby. $300 will get you into lot 2B and $95 will get you into lots 15/16.
Here's a map I made featuring parking lots and gates; parking lots are red markers and gates are green.
Residents near Churchill Downs apparently charge people to park on their lawns but I haven't seen this in action and I have no idea what they charge. I honestly think parking at Papa John's Stadium is the way to go. It's cheap and the walk isn't bad at all. If you can't handle the walk, you can do the shuttles.
Valet parkingAnother option is to do valet parking through Derby Experiences. Their lot is 1.5 blocks from Churchill Downs. According to their website, they charge $325 per person. They're parking cars, not people so I'm not sure why it says per person. It looks like even after paying $325, you still have to walk 1.5 blocks. For that price, I want Zenyatta herself to carry me to the entrance of Churchill Downs and a shirtless Mike Smith to escort me to my seat.
TaxisI've never used taxis but word on the street is getting taxi after the Derby is over is a 2-3 hour affair. After a long day on your feet, it's not something I'd want to do. A friend of mine once stayed in Sellersburg, IN and found a local to drive him back and forth. I assume the front desk clerk recommended someone to drive them. The driver dropped them off as close as road closures allowed and they had a designated pick up time and place. That sounds a lot more convenient than waiting in the taxi line for 2 hours. I'd try this instead before doing a taxi. It might be something you can use CraigsList for? I'll check when the Derby gets closer.
As usual, if you have any other ideas that I haven't listed, let me know!