You have a couple of options.
The first is to pick up a rental car at the airport from either
Budget, Dollar,
Avis, National, or
Alamo. You can then return the car to:
The Doubletree Suites at Downtown Disney (if you rent from Budget)
The Courtyard by Marriott at Downtown Disney (if you rent from Dollar)
The Hilton at Downtown Disney (if you rent from Avis)
The Car Care Center (if you rent from Alamo or National - the car care center is located just outside the Magic Kingdom parking lot)
You'll want to check with the rental company about their policies. I know Budget lets you return someplace other than the location you rented with no penalty, but I don't know about the other companies.
The pros to this are that you only have to pay for a shuttle one way to get back to the airport. There are also shuttles from the Downtown Disney hotels to get to the parks, so you won't have to worry about getting transportation from the hotel. (And if they check for a hotel ID, which they never did when I was there, Downtown Disney is a short walk away)
The con to this is that you'll have to have the car at the beginning of your trip, whether that's when you wanted it or not.
Your other option is to get a taxi or shuttle to those above locations.
The pros to this are that you'll be able to have the car when you want it. Also, contrary to Michael's experience, I've always had cheaper rates renting someplace other than the airport.
The cons are that you'll have to pay in order to get to whichever location you need. (Except the Downtown Disney hotels....you can easily take a bus to the market place and walk across the street)