動物病院って英語でなんて言うの？

We can abbreviate veterinary clinic' to 'vets'.

ie, I'm taking my cat to the vets tomorrow.

An animal hospital or veterinary hospital is a place where people's animalsmay have emergency care - just like a hospital is a place for emergency care for people. A 'vets' is equivalent to a 'doctors' or doctor's surgery - where animals are first taken for minor problems or diagnosis.
"Sorry, I will be absent from work this morning as my dog has to be cllected from the animal hospital."

The term we use is 'veterinary hospital'.
We may however say 'animal hospital' sometimes, most likely if talking to a child.

The shortened term is Vets.
You may hear someone say 'I need to take my dog to the vets'.

Veterinary clinic

This is a formal and very common way to say an animal hospital. You can also say "Vet clinic."

Veterinary hospital

This is another way to say Veterinary clinic, but implies that it is a more expansive building that offers more care. You can also say "Vet hospital."

Vet

This is the most simple and common way that a native speaker would refer to a Veterinary clinic.

Example:

You: "I found this cat and he looks sick."
Person: "You should probably take him to the vet"

veterinary clinic - a hospital for animals
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vet - same as veterinary clinic
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veterinarian - an animal doctor

Veterinary clinic
This is a hospital for animals
__
Example
A: My dog hurt his leg. Where do I take him?
B: You should take him to the veterinary clinic.

Anyone of the above would be suitable to use :-) I hope this helps!

Feel free to use any of them, they are appropriate to use for your question.