This is from memory here, so bear with me....
Frankly, in MK, most of the rides are family friendly:
Buzz Lightyear, Pooh, Small World, Snow white, Carousel, Dumbo, Train, Jungle Book, Country Bear, Peter Pan, Aladin's carpets, Tiki Room, Haunted Mansion, Riverboat, Carousel of Progress all would work with a little one.
EPCOT--Though more non-ride oriented, and therefore just fine for babies, the rides I can think of for little ones are The ride in the Land, The Ellen/Dinosaur ride, The ride in the Mexican pavillion (El Rio de Tiempo?) and the ride in the Living Seas.
I'm not as familiar with the other parks, but most of the other parks aren't "rides" except the "big" ones you'd have to do swaps with, as I can recall.
If you don't have one, invest in a sling. I had used a snugli when mine little ones were really little, but when we went when DD was 13 months, I found the sling to be sooo useful. Unlike the snugli, she was able to actually lay down crossways across my body and she slept alot and it wasn't as rough on my back.