The restaurants AND pool/beach waiters at the Royal Resorts are very kid-friendly (except, as noted, El Conquistador). Note that drink refills are usually NOT free as they are in the States. There is a kids' menu always available, and little ones are half-price for Mexican night and buffets. Also as noted above, the Express is great!
Many years ago when our one-year-old was fussy at dinner, the waiter scooped him up (with our ok) and took him on a tour of the restaurant and kitchen so we could enjoy the rest of our dinner.
At Happy Hour poolside or at the beach, you can order the kids inexpensive "fluffy" drinks without alcohol, and ask for chips, which will be delivered at no extra charge. My kids' favorites are Chocolate Monkeys, Piña Coladas, Banana Coladas, and Strawberry Daiquiris. Be careful to taste the drink first - occasionally they mess up on the "no alcohol" part, which I've found out when my kids said the drink tasted funny. Twice I thought it was ok but I guess I was wrong - they drank it and slept the rest of the day.