We went when my dd was 5 1/2 mo.....exclusively nursing. I've never been in a baby care center....but I hear they have very nice facilities for nursing moms. There are many nice quiet spots in the parks to sit down and nurse.....once you're used to it, no one seems to notice that baby is eating and not sleeping.
Personally, I am a very "on the go" nursing mom....my dd was usually in a babysling and I had perfected nursing while walking and carrying her by the time of our trip. With the sling and nursing shirts, no one knew she wasn't just sleeping. And she nursed a lot less during the day while in WDW and then nursed all night long! It was a good thing I'm able to sleep while nursing sidelying!
Are you looking for help with nursing in public discreetly? Or looking just for suggestions of where you can go for a little privacy?
Otherwise I found it wonderfully easy to take a baby. My 5yo used our stroller most of the trip and I carried the baby....she slept or watched what was going on. We just stopped to change a diaper occasionally and that was it! Very, very easy!