Whether you're visiting Cozumel for one day (cruise ship) or vacationing there for a week, you should definitely take a tour of the island. The best way to do this is to rent a "moke" for about $45; you can't get lost, as there's only one road.
Head out early to the Chankanaab Lagoon for snorkeling -- you have to beat the cruise ship passengers who arrive by 11am and get the water very crowded.
From there, head east to the "wild" side of the island, where there are no hotels and lots of rough surf -- you'll have a beach to yourself and a few rustic bars, offering grilled foods, cold cervezas, and great margaritas. You'll also find a few vendors that encourage you to bring home a souvenir.
If you are interested, there are small Mayan ruins before you get to the other coast. They're only a short walk into the jungle and guides are available -- for a small fee -- to give you a tour of San Gervasio.
As you round the corner to meet the ocean you'll find "Mescolitos." Stop in for the best margaritas and relax in the hammocks. But don't have a second drink -- you'll never leave!!
Continue on . . . there are a few more bars where you can enjoy an open-air luncheon by the sea. Then it's time for a swim and jumping the waves.
As you continue south, a visit to the lighthouse at the end of the island feels like you're on the earth's last tip of land.
If you want more beach time, you can head to San Francisco Beach on the southwest coast, where you'll have almost made the full circle. This is the best beach on the island, with nice white sand and calm water.
There are two Beach Clubs in this area that have sailboats, kayaks, wave runners, trampolines, umbrellas, lounge chairs, food, and booze. The cruise ship passengers usually leave about 2pm. The rental clubs close at 4pm.
If you prefer to shop, head on up to town for a couple of hours. Some unique gift, trinket, and jewelry stores await (as do the usual timeshare sales people and tour guides), all of them overlooking the malecon and pier where the ferry leaves for Playa del Carmen.
A moke is a 24-hour rental . . . so you still can use it to get out to dinner and probably another early swim in the morning.