I think the tram is your best bet. Hiking from Palm Springs is tough and HOT without much shade. If you have a car available, you can drive around to the Idyllwild side and go up one of the trails, but most of those have a good amount of gain.
The tram ride is a little expensive ($21.95 -- kids under 3 are free), but the ride in and of itself is worth it. There's some video of the ride at
http://www.pstramway.com/tickets-gifts.asp. When you get out of the tram, you're at 8000 or so feet in shady pine forest. You have to walk down a long concrete ramp, but when you reach the bottom, there are a series of fairly easy trails you can take. Probably the most frequent destination is Round Valley. There are a couple of steep spots and one spot where you have to kind of squeeze through/clamber over some rocks. I think it would be doable with a baby carrier; it would probably be easier if you have another adult with you in case you wanted to pass the child & carrier through/over something at any given spot.
The state park has some info at:
http://www.parks.ca.gov/?page_id=636 including a map at
http://www.parks.ca.gov/pages/636/files ... oSPmap.pdf There's a 4.4 mile loop you can do to Round Valley, returning via a route a bit further south. You could also call the Ranger station and ask them for recommendations.