Seeing as how it's your first cruise, my recommendation would be to book excursions through the cruse line, as you are not familiar with the ports.
I don't know about Carnival, but on Royal Caribbean, you have to be 21 to drink, and at least 1 person in each stateroom must be 21. Ages 18-20 were allowed to walk through a bar or lounge only if they had to in order to get to a theater where a show was.
In Cozumel, I have heard good recommendations about Paradise Beach. If you use the search feature in the "Caribbean Ports of Call" section, you can probably find information on Paradise Beach. It's easy to get to by taxi, and there are plenty of them at the piers in Cozumel. I haven't been to Progreso, so I can't offer any suggestions on there.
2007 - Majesty of the Seas (Bahamas)
2009 - Enchantment of the Seas (Western Caribbean)
2010 - Freedom of the Seas (Western Caribbean)
2013 - Golden Princess (Alaska, roundtrip from Seattle)