Seems you've gotten a lot of advice on rides. My daughter has been going since she was 2 months old, and was almost the size of a newborn, and we've never had any issues. At age 7 she has been on every ride except Maliboomer (and tower of Terror at DCA, but has been on TOT at WDW)
My number 1 recomendation would be the baby care centers. I like the one at DL better than DCA, but that's because I'm used to it more. They are climate controlled, the changing tables are wiped down often (when it's not too busy it's between every person, it there's a long line, sometimes the butcher paper just gets changed, and the table gets wiped between every other. They have little pottys for the kids training, high chairs, they will heat up food for you, a place for nursing, and they sell lots of different items you may need (extra food, formula, diapers, bottles, pacifiers, sun screen, etc.)
Unless it was an absolute emergency we would generally trek to the baby care center, I just prefered it over the regular bathrooms with the metal areas.
The one in DL is located behind the Corn dog cart on the way to first aid and child lost and found. Turn at the Camera store from mainstreet.
The one at DCA is located next to first aid behind the mission tortilla factory, that one also has a companion bathroom. I would usually make my pit stops while I was here with my daughter, just made it easier. They also have a table with some books. The one at DL sometimes has books out and sometimes doesn't but you can ask.