There are people who visit Cozumel far more often than I do--my last planned trip was supposed to be a month after Wilma hit (didn't have trip insurance and didn't need it--flights were canceled and hotel was closed). But I'd say go ahead and rent a car one afternoon to check out the other side of the island. It's been a long, long time since we've done it, and I don't think that having a drink at Coconuts or Mescalito's or wherever is so fabulous that it needs to be done every trip, but it's nice to get a feel for more of the island, especially as a first-timer. But if you're planning a mainland adventure, it makes it somehow less necessary. I don't think that driving on Cozumel or along the Yucatan coast is bad at all.
My main suggestion is that if you decide to see the ruins of Tulum, try to avoid the huge tour-bus crowds. First hour that they open has been good for us, after which we typically flee in horror. Really late might be ok too. Coba is in most ways more interesting and only about another 45 minutes or so inland, but that might be more of a commitment than you want to make, certainly if you want to dive a cenote (also a great experience). We once visited Coba, Chizen Itza and a festival in Vailladolid all in one day, driving out of Tulum. Don't even consider such a thing!
If you decide to more or less retrace Mitchsea's steps, and you find yourself hungry for lunch after your cenote dives, there are plenty of restaurants in the town of Tulum and at the little beachfront hotels a bit south of the ruins. I would assume there's stuff to eat (I only remember refreshments, but I assume there's a cafe) at the big, modern, shop-filled, touristy visitor's center too. Both the town and beach areas are a quick cab ride (many of the beach hotels are actually really close, but once they built the big visitor complex in the early or mid '90s, I lost track of whether or not it was still possible to shortcut along the beach road). At the beach, we used to sometimes stay at a rough little backpacker place called Don Armando's--it's called Zazil-kin now, and it's one of the first properties south of the ruins. I haven't been there in five years. If it's still about the same (and it appears to be in pictures I've seen), the food/restaurant/bar was pretty basic (and the accommodations primitive), but the beach is spectacular and has a view of the ruins if you're looking to cool your heels for a couple of hours in such a manner.
And if you find yourself in Playa del Carmen around dinner time, I'd consider dining there and doing a small bit of wandering before catching the ferry back, if just for the change of scenery. It's a pretty huge, lively resort town these days. Our first time there, there were still a few chickens running on dirt streets directly behind the beaches. It wasn't that long ago, really.