We've been taking DS since he was 7 1/2 months, and here are a few of our favorite spots.
MK - there is a play area for smaller kids next to Splash Mountain under the railroad, and Dumbo has an indoor playground to play while you wait in line that has a small toddler section in the center.
DHS - that's a tough one since the Honey I Shrunk the Kids playground closed. The Disney Jr. show has everyone sitting on the ground and then the kids getting up and moving around at certain parts. If anyone in your party is riding the Tower of Terror, use rider swap - if you go through the exit and through the gift shop, there is a space after that with benches that looks like an old hotel lobby. It can get loud/busy in the center when groups are exiting (which is constant), but off to the sides there is a bit of room.
Epcot - Lots of spots here. After (or through the exit) of Journey Into Imagination, there is a carpeted area with squares that make the noises of different instruments when you step of them. Also, if you go through the exit of Test Track, there is a nice, large air conditioned area where they have some cars on display. It's carpeted and never really gets too busy. The aquarium after The Seas with Nemo (again, you can go through the exit if she doesn't like dark rides) has a lot of room to move around, but the second-floor observation area can get pretty busy. In the World Showcase, walk back through the UK pavilion, and there's a garden area and hedge maze. Sometimes the maze can be busy with bigger kids running through, but the area right around it has tons of room.
AK- The Boneyard! Don't skip the Boneyard playground! DS actually started walking there, and it was perfect b/c the ground is that rubberized texture. There are some nice open spaces in there.
And, as just about everyone mentioned, the Baby Care Centers are awesome! Also, use rider swap (especially during nap time), and they do sell those cute Mickey Ear Hats that you can get names embroidered on in infant sizes...