Too many places to visit to list. I like the east beaches on Cozumel - just a few bars w/food and almost no people. If you are going to be there for more than a day, rent a jeep or car instead of taking the overpriced trips sponsored by the hotels. Also change your money to pesos to get better deals and never use a debit card - credit cards are usually OK.
Rent a jeep for several days or a week, however long you are there. It's cheaper than the taxis. Take a trip around the back of the island. Stay away from the west side day clubs. go to the Celerian light house. Eat dinner at Casa Mission resturante. anachesbbeachesbars.an.bbeachesbeaches