We've gone just about every route for this: nice big stroller, umbrella stroller, and using Disney strollers. I think for the 3 year old, the umbrella stroller may work just fine, but for the 20 month old especially, you're going to want a stroller that will let them sleep comfortably. Plus, this will give you a place to store diapers (if you still need them), snacks, water, etc. Just remember, like everyone else has said, you'll need one that collapses easily to take on Disney Transportation. With my first child we had a stroller that collapsed into a nice compact package, unfortunately the mechanism was a little complicated so I ended up snapping it trying to hurry to a tram. Now we have one that collapses with just a twist of the wrist. As kids get a little older, I think the umbrella is easier to deal with, but with the ages you're going with the convenience of having a comfortable place for them to rest outways the little bit of time you'll loose getting on Disney Transportation.