I've been to santorini a few years ago but that was pre-babies so can't really help you with anything baby-related.
i'd have to disagree with the pp, I absolutely loved the place. Yes, no rgeat beaches, mainly because half the island was a volcano and the signs are still there. However culturally it's absolutely brilliant. You can take a boat ride onto the volcano, observe all the different rock formations and the breathtaking views. There's a place on the island that is supposed to have the most beautiful sunset ever and a lot of people gather there to see it, didn't do much for me. And it there are also the ruins of a palace thousands and thousands of years old that is kind of still standing, was ruined by the volcano, definatelly worth to spend a morning there.
BTW, the main town thre is caller Fira, Thira is another name for the entire island.
I'm trying to remember, I think our guide said that Thira was the name of the island before half of it sank... not really sure, it's been several years.
We stayed in a village about 20'bus-ride from Fira and it was lovely and quiet, very personal service. The beach was nothing to write home about but it was clean with clear water. Sorry, can't remember name of village, if you want I can pull old photos out and try to refresh my memory.
May isn't particularly busy anyway. The whole island is a big hill, and especially in Fira it would be very difficult to push a pushchair around. Don't forget to check out the archeological museum in Fira, it's simply shocking how old some of the htings in there are.
Sorry I've rambled on, it's only cause I loved the place.
Take care x