I haven't been there with a 2 month old, but I did take my son when he was 6 months old. I was surprised at how easy things were. We brought our own stroller-a nice one where he could sleep-not an umbrella. He just napped in the stroller. You have to bring your own stroller because a baby that age can't use the rental strollers. We were staying off site with extended family. Because we got a late start to the parks, we ended up staying most of the day. I had planned on taking the baby back to the hotel, but since he slept so well in the stroller I never did. The baby and I, or dad, did take more breaks since baby couldn't go on things that his cousins went on.
The only rides I remember taking the baby on were Small World and some stuff at Epcot, like Illuminations. My husband took him on some other things like Pooh's Adventures, but really the baby just liked walking around and seeing all the people and things.
We also brought a baby bjorn (front carrier) for a change of pace. We went in December. In July I would be most concerned about the heat and dehydration. I would recommend that you ask your pediatrician about this. My dr. told us to make sure that we used bottled or boiled water for the formula, but not to give the baby any plain water (he was underweight). The only real problem with the trip was that the baby had an ear infection and the flying made that miserable for him. Are you flying? If so, make sure baby is nursing, or has a bottle/pacifier, during take-off and landing.
Hope this helps.