Great advice was already given by LongLiveRafiki!
I went in October with my 2 just-one-year-old granddaughters. DH, myself, my 4 kids in their 20s, 2 DILs, 1 SIL (my youngest DS is 22 and single). We stayed onsite at AKV and had 2 villas and used the buses. Luckily the weather was cooler, so we didn't have the heat/sun issue. Each park has a mom/baby station where you can feed, diaper, get a bottled water and have quiet time for a bit. One of my DIL's took advantage of this as she was still nursing. If you forget something, I think they provide things like diapers and wipes.
I created touring plans on the website with all the attractions we wanted to see but putting the pace as 'relaxed' and allowing for unexpected delays. It worked very well - I printed them out and had them with us.
In 1983, DH and I took our kids for the first time when they were 2, 4, 6 and I was pregnant with the 4th. Now, my oldest two are the dads
We always did the parks from morning to nights but this time we planned for early mornings followed by a flexible/later start the next day. It was the first trip where we actually sat on the grass in parks! In Epcot, we sat over by the coffee stand/Ellen's Universe of Energy (now gone) and in Magic Kingdom, it was across from Crystal Palace in the fenced in area. The girls enjoyed walking/crawling and looking around.
You might be surprised at what startles/scares the little ones. Some like the characters and some are afraid. They both loved the opening show at the Magic Kingdom. We watched
Disneyland Fun dvd many times before visiting and, after visiting they still can't get enough now that they saw the characters! We were lucky to stumble across Mickey in Epcot (with no line!) and they were so excited! One granddaughter cried her eyes out as he walked away. We had dinner at the Crystal Palace with Pooh and friends (our only TS meal) and they loved that, though one was a little shy with the characters and one was afraid of Eeyore.
Fireworks were not in the plans for the babies, so they left early. But one DIL had earphones? or something to quiet the noise and they stayed one night and my granddaughter seemed okay...maybe she was asleep before they stated, actually, I can't remember. Otherwise, I advise to leave the park before the booms!
They didn't like the indoor, dark rides as much - only one set of parents took their baby on the Haunted Mansion and both went on Pirates and FEA. They enjoyed Small World (of course!), the Carousel, the Living Seas, Jungle Cruise, People Mover, ... We split up a bit, but went to the parks together. We didn't go to HS this trip. We did MK, AK and E. Festival of the Lion King was a favorite! Sometimes they would sleep through things they probably would have enjoyed. They loved seeing the balloons. We used baby swap once we realized how it works. I'm a ride wimp and I brought a collapsible double stroller, so I would walk with one or both at times while the others enjoyed rides. The parents "wore" their babies if they weren't in my stroller. I brought a bug/sun netting for the stroller and that provided some darkness for napping, too. I also had a vinyl cover for rain but we didn't need that. I recommend a clip-on stroller fan. A nice umbrella stroller that is light but maybe reclines a little, has a little storage space and with a sun visor.
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We did not go back to the resort for a rest - the girls napped whenever/wherever - I do not recommend trying to plan that out. We got to the park early and stayed until we felt like leaving. Then, the next day we got to the park as early as we could but knowing it could be later and stayed until we felt like it. DH and I stayed later than the parents with the little ones sometimes. We did 2 days at 3 parks, so we had an early and late day for each and covered just about all we wanted.
Have fun!