Hi A.!
I agree with the other posters that having your "regular" stroller is the best. We went last January with our daughters. 18 months and 3 years old. We have a LARGE double, side by side jogging stroller. We took it with us. It was a little cumbersome to check at the gate and to ride on the buses around Disney. However, I would not go without it. Our oldest was in it more than my younger one. And our youngest could sleep in it whenever she wished.
We did go back to the hotel for naps. I did make sure that I had plenty of disposable sippy cups - you can get them at Target & Walmart. I also had a baggies of finger foods, diapers, extra wipes - not only for diaper changes but for clean up at meals. Our youngest was still taking a bottle, so for the afternoon time, I would pack a bottle in an insulated sleeve for later.
There is A LOT of good information on the following website. www.allearsnet.com There is a page specifically on traveling with toddlers. I found a lot of good information there. They also have a tips database that you can search as well.
Here are some of my tips.
At Magic Kingdom, if your child needs to "run" off some steam, there are a few places they can do this without a lot of trouble. As you are walking toward Cinderella's castle from Main Street, to the right, is a walk/path down to a covered area that used to be the start line for 20,000 leagues ride. It is a flat concrete area with an overhang. This is a perfect spot to let your little one roam on their own for a while.
Also, back in Fantasy Land, I believe, is Pooh's Corner. It is a playground area with a small cottage/house and pooh characters. There is a jungle gym like structure, but good, confined space for roaming and playing. Our girls did not want to leave. :)
If you haven't made your dinner reservations yet, do so now. You will want to make sure that you will be able to eat where and when you want without having to wait. Your toddler won't want to wait either. :) I highly recommend the Crystal Palace character meal at the Magic Kingdom (Pooh & Friends) or Playhouse Disney Character meal at Hollywood & Vine at MGM Studios.
There are baby care centers in all of the parks. From what I understand, they are AWESOME!! We didn't use them, but I have heard great things about them - especially if it is warm.
I have other tips as well, but do not want to manipulate your post. So, if you need any advice/tips with anything in particular, just let me know. I would be more than happy to help.
Have fun!!!
K. O