Many fall somewhere in between with a small admission fee (equivalent of $5-$10), perhaps a changing area and maybe some lights or ropes. Other than that, it's just a natural swimming hole. Some go underground for hundreds of meters or more, and they are a popular attraction for intermediate divers.
They're also perfect for just plain swimming. The Yucatan is hot and sticky; the water in cenotes is fresh and ice cold since they are partially or completely underground. In some cases, sunlight shines through holes in the cavern ceiling, illuminating the beautifully clear, blue-tinted water giving it an otherworldly feel.
And many cenotes attract swallows, which you might -- as I did -- mistake for bats. They constantly fly in and out squeaking busily. Though the clear water might trick your brain into thinking you're in a sterile pool, you are not. There is wildlife present, mostly in the form of little fish that will nibble at your feet if you are still, as well as the occasional turtle or snake.
Some recommended cenotes for swimming:
X'kekén near Valladolid is a beautiful underground spot. Cristalino off the main highway near Playa del Carmen is mostly exposed and has a fun cliff you can jump off (it's only a 15 foot or so drop). Grand Cenote near Tulum has some exposed and some subterranean areas and lots of wildlife running or swimming around.
So whether you go to Mexico for cave swimming and ancient sites or for mole and margaritas, it'll be worth it.
Confia en mi -- trust me on that.