Best Hotel

Well there aren't a ton close to both unless you're willing to pay an arm and a leg to stay at a monorail resort (Grand Floridian, Ploynesian, Contemporary) that will take you to both parks. There are also hotels like the Yacht & Beach Club and Boardwalk that are walking distance to Epcot and only a 10 minute boat ride to Hollywood Studios

They (MK and Epcot) aren't close together. Come up with more specific criteria.

about a 15 minute ride using any kind of transportation and also must have a pool

Contemporary, Poly or Grand Floridian ... All three are excellent and all three are linked to MK and EPCOT via monorail, boat or bus.

The hotels on the monorail route are not cheap, but they'll get you to MK or EPCOT in 15 minutes. The bus takes longer. These hotels have pools with slides.

Can I rent a car and go from park to park? How long would this take?

I'm not sure about cars, but when I stayed at the Animal Kingdom Lodge, we used the bus to get everywhere.

I know this is already stated above, but the three hotels that are connected via monorail to EPCOT and Magic Kingdom are Grand Floridian, Polynesian, and Contemporary. I've stayed at both the Grand Floridian and the Polynesian, and I prefer the Grand Floridian by far. The only thing the Polynesian hotel has over the Grand Floridian is the pool.

I personally recommend the Animal Kingdom Lodge. It is cheaper than the Grand Floridian, Polynesian, and Contemporary, but I found it to be the best by far out of the three resorts I've stayed at-Animal Kingdom, Polynesian, Floridian. Although you do have to take the bus everywhere, the food is amazing, the wildlife is awesome (even if you don't have an animal view room, you can still see the animals from the lobby), and it has amazing theming.

I recommend the Dolphin near Epcot. It's not a full-fledged a Disney resort (it's owned by Starwood), but it offers all the Disney transportation with the added benefit of being one of the boat stops in the lagoon between Epcot and DSH. In my experience it's just as nice as Disney's "moderate" resorts and usually a little bit more affordable.

"Can I rent a car and go from park to park? How long would this take?"

The first question is obviously yes. The second question depends on how fast you can drive, how well you know the roads, how full the parking lots are, and the distance you walk to the main gate. I bet it will take at least 30 minutes driving from a moderate or value Disney resort. It's less convenient due to the walking and might require a tram ride. NOT a good idea if you're going to the Magic Kingdom. Just take the bus. You don't want to be bogged down at TTC.

Blake, why don't you look at air photos of Walt Disney World. Get your bearings that way and maybe we can fine-tune your selection options.

I am going to recommend Carribean Beach. Nice, but middle priced!

If you want the whole package, I would recommend the Beach Club

Any of the EPCOT Resorts hotels are my favorite. Easy walk to World Showcase, and although you can certainly take a boat to DHS, I personally just walk, which takes about 20-25 minutes, but it's a fairly nice walk until you need to navigate the parking lot near the entrance, anyway.

Also, just walk through EPCOT and catch the monorail to TTC and then get on the MK line.

You can hit 3 of the 4 parks without getting on a bus, and to me, that can't be beat.

For the experience, getting an animal-view room at AK Lodge is a must-do. But it's still inconvenient to get to parks.

I've been thinking maybe the Swan. I heard that it wasn't an official Disney resort and therefore, didn't offer some amenities.