We took our kids to both Disney Land and Disney World when they were little - probably starting when the youngest was 15 - 18 months and in a stroller. Typically this was when I was attending a business conference held at the resort, or we were all visiting the elderly retired grandparents living in Florida - that is, we were adding the theme park to a trip we were making anyway.
I prefer Disney World, staying at one of the hotels on the monorail. That way it is a short trip back to the hotel for a nap, lunch or just a break for swimming in the hotel pool, which is very helpful for the youngest kids, less than 5 - 6 years old, who tire more easily and still need naps. I second the vote for the character breakfast - the kids loved that! There are also multiple parks in both the Disney World complex, and in the area. My kids enjoyed visiting Sea World and the Kennedy Space Center, and also Daytona Beach, which is about an hour from Orlando; they loved the fact that you can drive your car on the beach itself.