I'm the family cruise editor here and have written an article on this section titled "Bringing the Baby" which answers many of your questions. Here are answers to some that were not addressed in the article:
*Babysitting: Carnival does not accept those under two years in the free youth program. However, they do have limited group babysitting for infants and toddlers usually in the morning on days in port and also after 10 p.m. most nights. There is a fee for this and the counselors will change diapers. Check in with the youth counselors in the kids' room early on during your cruise for specific details.
* Parmalat milk: Although you can get milk at the buffets and restaurants, you may want some milk to have in the room if it's needed during the night, early morning, etc. I live in NJ and we can get "parmalat milk" which comes in cardboard containers and doesn't have to be refrigerated. We can get it at our grocery store and also at health food supermarkets (Whole Foods). If you bring a few cartons, pack them in your carry on -- once they burst in our suitcases we checked in.
* Climate: It should be pretty warm in Mexico -- pack summer type clothes but also windbreakers for evenings.
* Medicine: Bring any typical over the counter pediatric medicine you may use such as tylenol, teething gel, etc. It's extremely difficult to find pediatric medicines once you're under way.
If the above and the article hasn't addressed your questions, feel free to get back in touch with me.
Luisa Frey Gaynor