Cheeky in TN, first off, welcome to the boards!!!
As for the little one, there are actually many rides that you can take him/her on. We took our DS to the World, for our annual family gathering, when he was 4 mo old and had a blast. In the Magic Kingdom, most all of the rides in Fantasyland are suitable, along with the Jungle Cruise, Buzz, and my all time favorite with an infant, the People Mover (TTA) - a great way to cool off with no line! *Note: I would take caution when it comes to Pirates of the Caribbean and Haunted Mansion. Obviously, you know how your child would react to these better than anyone; however, I say this only because of stories I have heard from friends and their youngsters.

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At MGM and AK, shows are always a great escape from the heat and an easy exit is always available if the baby is fussy or scared - we usually entered the theater and then let others sit so we could be close to the end of a row. I would again suggest you be cautious with shows that have a lot of noise (ie: Tarzan); however, shows indoors such as The Little Mermaid, DS thoroughly enjoyed - great lights and music he loved
You are smart making your way back to your room early in the day. Get a break from the heat and let the baby nap in a quiet room. This was the best thing we could have done. Our DS always napped for a couple of hours and was refreshed and ready to go mid to late afternoon.
Best of luck and have a magical time!
