There are two clinics on the island that have English speaking doctors and a good level of care. The small Clinica San Miguel on Calle 6 Norte between Ave 5 and 10 is favored by many as it tends to be a bit cheaper than the other one, the CMC which is located on Calle 1 Sur with Avenida 50.
No one goes to the government hospitals unless they are desperate. For major surgeries or speciality items like vein removal, for example, hernias, that kind of thing, it's sometimes worth it to go to Cancun. You can fly over cheaply via Maya Air. And for serious stuff at very affordable prices try Star Medica in Merida which is a big center for international medical tourism and has some of the finest doctors in the world, I'm told.
If you are local and can prove it with a card, you should be given discounts at the Clinica San Miguel and the CMC. I am told that both take US insurance also but best check on that.
As to dentists, I never go to anyone except Luis Erives who has offices one storefront in from the intersection of Avenida 65 and Calle 11, northeast corner. He does really excellent work at affordable prices. Had numerous crowns done by him and he also removed several of my daughter's wisdom teeth.