We just returned from a week long trip with a 3 month old (our first too) and it wasn't bad at all. If you go in with a relaxed attitude and realize that the trip will be different than an adults only trip, you won't have a problem. I think that it also helped that we had been there in Decembe when I was 6 months pregnant, so we were used to going at a slower pace.
We brought our own stroller from home. The park strollers really do not work for infants, they are more designed for toddlers. I would invest in a good, light stroller. It was a pain dragging it on and off the busses, but so nice to have for the walk back to the room. Also, most of the restaurants did not have infant seats, and Brady was too small for a highchair, so they would let us take the stroller into the restaurants since it wasn't too bulky.
I carried two extra bags into the parks. It was a pain to get through the bag checks, but once inside, they were shoved under the stroller and it wasn't bad. One bag had diapers, change of clothes, changing pad, wipes, hat, sunscreen and a very light blanket. The other bag had bib, burp cloth and bottles and cereal (he was on cereal at that point).
The baby care centers in the parks are great. We used them a lot for feeding and changing him. They have tons of nice changing tables and high chairs and also supplies to sell. We usually timed our day that he would get cereal and bottle before we left in morning, lunch bottle in the parks, bottle and cereal back at room in afternoon, dinner bottle in the parks, and bottle in room before bed.
The resorts have pack-n-plays for the babies so we didn't have to worry about where he slept. We washed him in the sink at the resort. We washed his bottles in the resort sink each night. It wasn't too bad at all, just took a little more time. We brought most of our stuff with us, but you can also use a grocery delivery service to deliver bulky things if you don't want to pack it.
My mom went with us, and it was nice to have the extra hands, especially on the busses. One person would take Brady, another the bags, and another the stroller. We used baby swap on the big rides, so we all got to ride everything and have someone to go on with. But, it is amazing how much an infant can do at Disney! Basically anything that doesn't have a height restriction, the baby can go on.
Here is a list of all the extra things we took this time:
Bottles
Bottle brushes and dish soap
small bowls and baby spoons
Canister of baby formula (we had bottled water delivered)
Box of baby cereal
Package of Diapers
Small package of wipes
Swim diapers
Baby sun block
Blanket for the baby
Baby clothes
Disposable baby washclothes (Huggies brand)
Stroller
Changing pad
Clip on Stroller fan (lifesaver! I bought on ebay and everyone commented on it)
Ziploc bags (gallon size, I found a ton of uses for these)
Anyway, I hope this helps. We had a great time and it really wasn't bad at all. We saw and did plenty and the baby had a great time and really enjoyed the colors and sounds. It takes a little planning and organization, but travelling with an infant isn't bad at all.