what is the place called where cars are sold

a car shop?

It depends if the cars are new: a car dealer, if they're second-hand, it could be a garage.

In the U.S. a 'car shop' would be a place where cars are fixed, e.g. I am taking my car to the shop - I am taking it to be repaired.

Cars, both new and used, are sold at a "car dealership". A small establishment which sells only used cars is sometimes called a "used car lot".

a dealership

dealership, dealers (-s on the end same as in butchers, bakers - it's debatable whether there should be an apostrophe with the s but you commonly see it without).
Also car showroom(s) is in quite common usage here in the UK.
You might also hear garage used as a general term, though that probably implies that repairs as well as sales are carried out there.

In the U.S. a 'car shop' would be a place where cars are fixed, e.g. I am taking my car to the shop - I am taking it to be repaired.

I don't know about that...I agree about the car being in the shop, but I have never heard the expression "car shop."

Also, garage is not used in the US for a used car dealership. A garage (or "the shop") is where a car is repaired.

Here (NZ) we say, car dealer, or car showroom for new cars. For used cars, car yard.

Garage or workshop for repairs. Panel beater for repairing dents.

Yep - as above - it NZ i would call it a car dealer or a car yard.

Or, a cheap way to get a car to drop someone off at the airport and not pay for a taxi....

RE #5,
I would not say Car Shop, but just The Shop to refer to the generic place I go to get my car fixed that is not The Dealer. If I am feeling rich and take my car back to the place that I bought it to get serviced, then I would say I'm getting my car serviced or fixed at the dealer. At this point everyone around you will moan and tell you about their favorite shop where you should take your car instead.

"My car is in the shop" would mean I'm having work done on it, and "the shop" I'm referring to could be called a "body shop." There are now gyms with the cutesy name "Body Shop," but that name has been taken from its earlier use to mean a place that did car repairs.

"Body shop" itself was used because some work done was not just a tune-ups or oil change, but work on the car body itself, such as the repair of dents, replacement of a damaged door, etc.

When I mentioned a "garage", I actually had in mind a gas station/repair place that also sometimes sells used cars.

If my car needed repairs, I would take it to a garage, not to a shop.

#9 -- It may be a regional thing, but on the east coast "My car is in the shop" might mean a body shop or might mean any kind of car repair facility. My car could be in the shop for work on the brakes or the transmission etc, not just body work.

Generally in the U.S., used car lot (even though, strictly speaking, that is a location and not a business) or used car dealer, would convey the meaning. In many cases, however, used cars are sold at the same dealer as new cars (just in a separate action and sometimes called "previously owned" cars or other variations to get away of the bad connotation of a used car). Used car salesman has long been used as a term for a dishonest salesman.

When I saw that a Bubba had replied, somehow I knew that it would be about used cars.

Yet, one of the famous dealerships in New Orleans is Bubba Oustalet. I grew up watching his annoying commercials on television.

VinnyD is correct at #11. My car might be in the "body shop" for body work, as it was after someone sideswiped me in a parking lot; it might have been at the local gas station, where there is a mechanic on duty for various other kinds of work. In either case, I would have said "in the shop," and my answer above was incomplete. It was not a matter of regionalisms.

My Hinglish says the car is being repaired at the car repair vala's( place)/gaadi maramat karnevala ki jagah/ the car man/the mechanic's etc.
I buy a car from the car yard and if it has a dent I take it to the panel beaters.I park a car inside the GARage( English pronunciation).
Many ozzies use the slang "servo" for service station where one buys petrol but no longer gets any service of a mechanical sort since the supermarkets bought up all the Petrol Pumps in the cities in Ozstriya.

#15 -- I'm not English but it seems to me I hear garAZH at least as often in the UK as GARridge. UKanians please correct.

i may not be the best indicator but i feel like i've never heard garAZH and that it would sound really weird (except from the queen's mouth maybe)

What kind of car shop are you talking about though? If it's the place from where one buys cars, in Malta we call it a "cars showroom" or just "a showroom" for short. 2nd hand cars are bought from an autodealer. We also take the car in for service to the mechanic's workshop and we park it in a garage (we pronounce the last "g" like the "s" in "television" and the second "a" like the "o" in "god" in American English).
Service stations are called Petrol Stations in Malta, even if they also sell diesel.