Gotcha. Well, my DDs first Dream experience was on the Dream at 18 months. We went with the early seating, b/c she usually eats between 5 and 6 at home. However, the long dinners usually meant myself or my husband was up and down from the table walking around with her at any given time past about 35-40 min. She LOVED Nemo's Reef (the splash area0, which is pretty much all that's open to untrained toddlers. But it didn't matter b/c she was in heaven in there. You do have to worry about bigger kids going in and running around a bit rambunctiously, but you can ask a pool guard to step in if it gets really out of hand (I usually only had to say something on the stingray slide).
If you're on the east coast, it will be difficult for your LOs to stay up for the shows if you have early seating, so my DD did not watch any shows that first trip. I'm on the fence about whether (in hindsight) I would have preferred just choosing the second seating and going to the first show, and then grabbing food at Cabana's or on deck. Still not sure. If your LOs are comfortable with sleeping wherever (mine was a poor sleeper, so we didn't bother), I know some people will take their kids to the nursery and enjoy dinner by themselves. I wanted to enjoy every moment as a family, so we skipped things my DD couldn't participate in.
At that age we didn't book any excursions, just enjoyed ourselves on board during Nassau. DD LOVED
Castaway Cay. We brought a float from home, which really helped her to enjoy the water (the pop up kind, that fold like car sun shades). We also bought the sand toys.
She loved the meet n' greets, but most of the books we read to her at that age were Disney or Princess themed, so she knew the characters going in. She loved exploring the ship and sitting in the porthole-type windows in the halls. We went into the Kids Club open houses so she could play in Andy's room and color.
You'll find people that disagree, b/c they want to take advantage of their time on board, but we made it a point to try to stick to her schedule as much as possible in terms of eating and nap/sleep time. Sleep times were usually off by about 30-60 min, but we still made sure she napped every day. The first day was the most difficult b/c the room wasn't ready in time for her nap, so we had some serious crankiness for the last 30 min or so. For naps we alternated who got to wander around the ship while the other stayed in the room. We also had a verandah so that helped.
That's what I can think of for now, if you have specific questions, ask away, and if I remember anything else, I'll add it.
It was the best vacation for our family. TAKE LOTS OF PICTURES!!! Our DD was very verbal at a young age, so she talked about the trip a lot when we got home and asked to go to the "Nemo fountain" and "Mickey boat" for the months that followed. And she liked looking at pictures of herself from the trip. It was truly magical.