Chen Rio is the most beautiful beach on Cozumel, in my opinion. It is on the east side of the island and has to be reached by either a car rental or a taxi. You can arrange to have a taxi take you in the morning and then come back for you in the afternoon, or simply just stay with you if you want to explore the island. Chen Rio offers a palapa restaurant with excellent seafood and a bar with reasonable prices. Since it is primarily a "local" beach bringing your own cooler with beer, soda or whatever is fine.
If you like tradtitional Mexican food rather than seafood, Coconuts restaurant is only about a mile away and is another unique experience. You can negotiate rates with the taxi drivers hourly or for the day, if you do not want to drive. A word of caution, however. I would not recommend renting a motor scooter on Cozumel.
The easiest way to get to Chen Rio is to take the cross island road from downtown and when you reach the east coast turn to the right and head south until you see the sign marking the entrance to the restaurant and beach. It is worth the trip.