The Spanish yellow pages lists 71 veterinary clinics and hospitals in Tenerife, so there should be one somewhere near you.
Check out this link for a Google map showing where they are.
When we were staying in Los Cristianos, the clinic we used was the one in Las Americas, Clinica Veterinaria Sur, in the El Camisón urbanization on Avenida Antonio Dominguez. It was when Max was being neutered (poor lad) and the staff there, although Spanish, all spoke good English.
Max was given excellent treatment and the vet and her assistants showed as much love for animals as anybody from any other country.
After we moved into our place at Guargacho, we used a vet in nearby Las Galletas.
This was Centro Veterinario Las Galletas, found on Calle La Paz in that village.
With staff seemingly of Scandinavian origin, this was an equally excellent clinic, with good English being spoken and treatment carried out expertly and with loving regard for the animal.
We used this vet when obtaining all the necessary documents to bring Max back to the UK with us and they were extremely helpful.
I would say that this is the standard you should expect from any veterinary clinic on the island.