Walk, Bike, Driving Yourself (free parking on site), Shuttle or Uber
Coachella does offer a shuttle that services all nearby areas stretching from Indio to Palm Springs. The shuttle runs throughout the day and costs $75 for all three days of the festival. You need to purchase the shuttle separately from your Coachella passes. You will receive a shuttle card with your tickets that must be shown to board the bus in either direction.
This is highly dependent on where you are staying for Coachella. If you are close enough to bike or walk I will always recommend going that route for the convenience and lack of traffic. If you aren't that close I highly recommend the official shuttle. They run very efficiently, are reasonably priced at $75 for the weekend, have pickup spots throughout the region and you can drink on them. You can drive yourself as there is free parking at Coachella or you can Uber, but the shuttle is far more convenient and is reasonably priced. It's our recommended way to get to Coachella daily.
If you can walk it would be free obviously, but most likely you will need another method. If you drive to Coachella they have free parking on site, so this is another inexpensive option. If you can afford it, the shuttle is our preferred method and will run you $75 for the whole weekend. Uber will quickly become more expensive than this, so go with one of the other methods if you are trying to keep costs reasonable.
Yes, Coachella offers free parking on site. You cannot leave your car overnight but it will be open daily for the festival.