Cozumel is a great place to snorkel. We went on a ship's excursion to the Palencaar Reef it was amazing!! If you want to just relax & chill out on the beach and maybe do a little snorkeling then i would suggest you hop a taxi to Channunkaub National Park - nice beach, souvenior shops, dolphin swim, mini-ruins, restaurants. If you are into history and ruins then a trip to Tulum would probably be a good idea. The shopping in Cozumel isn't that great, you can get some decent deals on gemstones.
The "standard" is beaching at Chankanaab, then wandering around San Miguel. Many people go to Coz just to dive (or snokel, or SNUBA) the Palancar Reef. Breathtaking.
Tulum is lovely, it's the only Mayan ruin facing on the ocean. I do not recommend it if the strait is rough and you tend toward seasickness. There's a jitney at Tulum so there's not a lot of walking. Lots of shopping.
When we were in Belize, we saw several big cruise ships anchored out. It was about a 45-minute run into Belize City from the ship. Each way. The "city" tour isn't worth it (IMHO) mostly just a shopping mall, every place selling the same thing.
I know Carnival does a trip to Bird Cay (we saw them there). Great snokeling (very shallow water) but I imagine it sells out fast. All transportation from cruise ships at Belize (that we could see) was done by local fast boat, not ship's boats.