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"Is there a single official C compiler, and how do I install it?"
No, there is no compiler provided by the C standard and thus no way to install it.
You have to use a third-party compiler. The most common for Linux and macOS are called GCC (GNU Compiler Collection) and Clang.
Since you ask for Windows, there are several C compilers available.
Take a look at the answers and comments to this question of mine a while ago:
Has Windows an integrated built-in C/C++ compiler package?
There you can find the at least many options.
The way to install you can find in the respective documentations and isn't object to be defined here as this would explode the frame.
My personal recommendation is GCC with the MingW_64 package. How you install it, you can find, as said, in the documentation.
Is there a single official C compiler? No.
Are you sure that you insist on the "official" one (which is not even clearly defined as the other comment pointed out)?
In your situation I would look for a "convenient" installation.
This more or less leads to the answer to "How to install it?"
The selection and indirectly the way to install it, is most wisely based on asking your friends/colleagues/fellows, because you will have questions and they are primarily best answered by any human you can directly ask. StackOverflow is only second place.
If that does not help you selecting one, I recommend to use yoru favorite search engine on "install gcc". In mine, the 10th result is my favorite.
Maybe go one step further towards "convenient" and search for "free IDE C" my favorite is the 6th result.
But of course this is not to tell you my favorite in a hidden way. You just look at the results, check the first few out and decide based on what you like best.