My son was 15 months old on our first trip to WDW. My mom had tried to get us to leave him with her & my dad so that we could, in her words, spend quality time there with our DD, then 4. We refused to leave him behind, and he had a wonderful time! Going at a cooler time of year is so smart. We went the week after Thanksgiving and the temps were wonderful - no sweating, hot, and miserable baby. He loved everything about the trip! He loved the rides, the parades, the shows, and most of all, the characters! The only characters he really "knew" before were Pooh & friends, so I booked a character meal at Crystal Palace. When he saw Pooh across the room, he started bouncing in his high chair, pointing & saying, "Pooh!" over & over. It was precious!
My son napped everyday in his stroller. I know some are advocates of going back to the room, but we had much shorter park hours. My son would fall asleep and then he & I would find a quieter spot to hang out for awhile while my husband took my daughter on some rides. Then we would switch out. My son also liked the toddler play areas in Toon Town. They didn't have the Pooh's play area then, but that would be perfect for a little one. They can get out of the stroller and run around a little. Oh, and we took our own stroller, so it was one he was used to. Plus, it reclined, making those naps more comfortable.
Dining wasn't a problem, either. We ate at several sit-down restaurants, including Jiko, the Hollywood Brown Derby, and some lower key places. All were very accommodating, offering my son fruit, which along with Cheerios was about all he'd eat at the time! We planned early dinners so that he and my daughter weren't overly tired.
My best advice is to go and enjoy your child at the most magical place on earth! Of course my son, now 8-1/2, doesn't remember that trip to WDW, but I certainly do. He had a wonderful time while there. And I have a heart full of wonderful memories of that first trip.