From reading port reviews we knew there wasn't much in Progreso beyond a trip to the Mayan ruins or shopping. We did neither. We took the free shuttle to town and then walked to the public beach. Nice and lively atmosphere as there were several dozen passengers from the ship with the same idea. We landed at a place called Tommy's on the beach (main building and bathrooms) across the street. The drinks and chis and salsa were very good and the service top notch. We enjoyed the warm weather and spent about an hour shopping in town before heading back to the ship for dinner and then the New Year's Eve party on Lido.
From reading port reviews, it sounded like we would have our choice of several private beach clubs on the western shore of Cozumel. We took a taxi to the Ocean Beach Club...it didn't look like much and the place was adjacent to other Beach Club. We walked next door to San Francisco club but the place next to it looked nicer still. We ended up at the Paradise Beach Club and felt lucky we had kept walking south along the beach. Beautiful club with lots of water inflatable toys, kayaks, slides, snorkeling, etc. Wave runner and para-sailing next door (separate contractor) if you wanted. Paradise also had a huge outdoor pool and bar behind the beach area where the water was clear and warmer then the gulf - plus the bathrooms were very modern and clean. Besides Carnival passengers, there were a lot of passengers from the Celebrity X Cruise ships docked in Cozumel too. Worth the $15 taxi ride each way form the cruise terminal, highly recommend.