What do you call the place where car tires are changed, oil is checked and if there is something wrong with your car, it is fixed there? I would like to know what people call that plece in American and British English.
The umbrella term for all the various types of specialist places you might take your car to for repair is a "garage".
My car is in the garage.
Yes, the outhouse on a home where you may store a car is also called a garage - it is all about context. I'm going to the store is always understood as being the place where you buy things when you use it in context, even though the word "store" can mean other things.
As has been mentioned in comments, there are specialist garages that deal with specific aspects of car repairs. In the UK, a garage that fixes dents and bumps in the exterior body is called either a body shop, or a coachworks, but these are still types of garage.