I've used the following blog post as a guideline for teaching my 1 year old how to behave with pets.
To summarize, teaching the child to respect the boundaries of the pet (instead of trying to actually play/care of the animal), will make the pet NOT afraid of the child. That in turn allows the pet to choose when to interact with the child. Turns out, at least in our case, the relationship between the child and cat in our case, is very similar to how the adults interact with the cat.
https://dogsandbabies.wordpress.com/2011/01/24/mamas-dont-let-your-babies-get-magnetized-to-dogs/