I live in Tampa, so my daughter who's 19 months has been 11 times. We have Florida Resident passes, so we don't go during the summer b/c we can't.
However, the first time I took my daughter, she was five months old. She had a blast! She also cut her first tooth on that trip.
If you do decide to bring your young one, remember plenty of water. Julie drank more water on that trip than I think she's drank her entire life. Lots of sunscreen and breaks are good too.
Ways to cool off .... there is the train ride at MK. Best way to get your little one to sleep to, if they're fighting it. Julie loves to snuggle on the train. If it's getting too miserable, we go into the shops, go back to the RV for a break and swimming.
Be prepared to carry the little one though, Juliette decided that she was NOT going to ride in the stroller on our second day of the trip. She's also a pro at skipping naps, so we rode the train and other slow rides a lot.
Also, most of the rides you can take the babies on. Julie has always LOVED Snow White, it's one of her favorites still. And, three times now she's fallen asleep while waiting in line for Pooh's Grand Adventure.
Oh, and the rain ... Babies R Us sells a rain cover for strollers. We weren't prepared and were at Epcot, so we got a Poncho and put it over the stroller since Julie was napping one day. When she woke up, I just put her in the poncho with me, she laughed during most of the day. Also, most of the time, the rain storms are no more than a few minutes ... you can always just dip inside for a meal or something ... it's doable.
My fellow Floridians are right, we do survive on A/C though. Imagine being pregnant down here during the summer and a lovely hurricane comes through and knocks out your power for a week. Now that was fun. After that week, I can withstand anything Disney can give me!
Amanda