We took our DS to WDW for the first time last fall right after he turned 1. It was fantastic! We let him set the pace since we had been many times before and had our chance to run from headliner to headliner. He napped in the stroller, and we fed him in the Baby Care Centers to get away from the chaos of the park and let him focus on eating instead of people watching. Also, a great place for him to run around and burn off energy since he had just learned to walk. He absolutely LOVED the characters! We got his first haircut done at the Harmony Barber Shop. He enjoyed all of the attractions, although we did find ourselves avoiding shows because he didn't want to sit for long periods of time. Although he mostly napped in the stroller at the parks, we still did get away back to the resort for an afternoon break. Even if he didn't nap, it was nice to relax a bit before heading back in the evening. We mostly had him back by his home bedtime of 7:30, but a couple of nights we changed him into his pajamas and let him sleep in the stroller so we could stay later for the fireworks. He would go right back to sleep when we got him back to the resort. I think for him, it was the PERFECT age to introduce him to Disney since we know we will be back many times to come.
Just last month, we took him to DL also. This time, he was about 19 months old. He was a lot more interested in the attractions and seemed to have a better time interacting on them. Lots more dancing, clapping, giggling, etc on the attractions. A little hesitant with the characters compared to our WDW trip, but he was not scared of them. Still a little too young to be scared during any of the attractions either. He didn't nap in the stroller, so we mostly stuck with his home napping schedule and went back to the resort to let him nap for a couple of hours before going back out in the evenings. He adapted well to staying out later, and even though he wasn't getting as much sleep at night as he was used to at home, he didn't get cranky on us. I found it was a little more difficult since he is used to walking and running around at home, so he didn't want to be in the stroller as much or held in the ride lines, he wanted to get down and explore. We tried to find opportunities to let him do this without disturbing other guests to keep him and us happy.