We took DD when she was 9 months. I definitely recommend the baby care centers, they are an oasis of calm. The one in
Disneyland is towards the end of Main Street, next to the Plaza Inn. The one at California Adventure is next to Carsland, near the Ghirardelli.
This was 3 years ago, but the procedure for rider swap was show the CM the baby, at least one person in your group needs to enter the line, then they will give you a rider swap pass. When you use the pass, you still go to the entrance and the CM will direct you to a different line (probably the Fastpass line). We even did this for Toy Story Mania, which doesn't have a height restriction, but the CM only gave it to us grudgingly after we showed him DD was asleep in the stroller.
Some rides that I recall being nice for DD were Small World, Jungle Cruise, the train (though she fell asleep), and Heimlick's Chew Chew Train. The ones that had no height restriction but still made me nervous were Toy Story Mania (jerky motions) and the swinging version of Mickey's Fun Wheel (we didn't know what we were getting into!).