I took my first Disney World trip with the kids when the youngest was 8 months old and my oldest was 3 y.o. and have been going back every year since. I loved my trips with the baby. Disney makes everything so easy. As long as you follow the child's routine, there should be no problem. I knew I would not be able to see and do everything but the slower pace made me enjoy all the little details that I would have missed if I was doing it commando style. Here are some tips that were greatly useful to me:
I highly suggest you to stay onsite. I don't know if you are flying but if you do Magic Express Buses are fabulous! The luggage being taken care of, all you have to do is board the bus and enjoy the ride to the resort. I always stayed in moderate resorts with the kids, my favourite being POFQ which is the smallest of all resorts with only one bus stop. The small size is perfect when traveling with a baby/toddler. It means you can get to the room quicker with minimal walking at the end of the day when everyone is tired. If you can afford it, than I prefer a monorail resort because you don't have to fold up the stroller and it offers the convenience of just rolling the sleeping child right onto the train.
There are Baby Care Centres in all the parks which have a changing area with changing tables, a feeding area with a sink, microwave, and highchairs, and a breast feeding area with rocking chairs. It even has a TV area where the kids can chill out and rest for a bit. You can also buy baby items that you might have forgotten to bring to the park. They carry everything from diapers, to baby food, to medicine.
There are also splash pads and water playgrounds around the parks where the young ones can run around and cool off. There are also tons of rides that are great for little ones so no need to worry about that. You will have plenty to do.
I suggest bringing a good quality umbrella stroller with a back rest that is completely retractable and a sun shade. They are lighter in weight, easier to manoeuvre and take a second to fold up compared to a regular stroller. This is especially useful for the bus rides back and forth from the resort to the parks. I also brought a rain cover and a sun cover for the stroller which came in very handy.
If you are flying down, I suggest you bring the child's clothes and only the essentials but you can order diapers, wipes and baby food from Garden Grocers and have it delivered to your resort before your arrival. This saved us tons of luggage space and those extra charges for baggage weight.
I also suggest to either go back to the resort in the afternoon for a little rest or pool time. If not, then I would book an ADR at a sit down restaurant so that you can get out of the hot afternoon sun and go somewhere more relaxing for the child.
If you want more tips or you want to read about my experience visit my blog:
http://disneyplaning101.blogspot.ca
Hope that helps.