We took our daughter right for 5 days after her first birthday and she loved it!
Yes, they still have parent swap, but we didn't use it. There are so many rides that she can go on and she'll love. Small World, Dumbo, Peter Pan, Pooh, Pirates, Fantasmic (she did get scared because it was so dark), Haunted Mansion, Jungle Cruise, etc (and that's just at MK!) We also took her for 1 day when she was 8 months old and the big difference was she's more aware of things that scare her now. Haunted Mansion at 8 months didn't phase her, but now it did. That's a kid by kid basis thing though
The biggest thing to remember is to go at her speed. We always went back to our resort in the afternoon so she could have a proper nap and we let her sleep later for her nap so she could stay out later at night. I know some parents are totally against this and try to keep their kid on the same schedule as they are on at home, but I think at Disney that's nearly impossible IMO. Again, every kid is different, and ours has no problem getting back on schedule once we were home.
We brought our own stroller and I really advocate that. First, its like $15 a day to rent one of theirs and they really don't look as comfortable as even a cheap umbrella looks.
Also if you go in the summer, remember to keep her hydrated and take advantage of the baby care centers with the awesome AC

. The good thing about most of these rides are that they aren't out in the open so you get a good mix of heat and AC to balance things out. Take breaks and sit in an air conditioned counter service place. I think there's a whole list on here of the ones with AC.
She LOVED the character meet and greets. Got such a kick out of everyone. I can see how some babies/toddlers would be scared, but luckily she wasn't
Biggest thing for me is to just remember that you're on her schedule when you're at the parks. There's gonna be stuff you don't get to because of her, and that's OK.