Coming from Python, the whole C/C++ thing is kind of alien to begin with... and then I see in one place that Arduino uses 'standard' C, and in another that it uses 'standard' C++, so on and so forth. Which is it? My admittedly crude understanding of the difference between the two is that C++ is (roughly) C with classes/objects. How does that affect which language or dialect (C or C++) should I concentrate on learning for use primarily with Arduino?
Arduino doesn't run either C or C++. It runs machine code compiled from either C, C++ or any other language that has a compiler for the Arduino instruction set.
C being a subset of C++, if Arduino can "run" C++ then it can "run" C.
If you don't already know C nor C++, you should probably start with C, just to get used to the whole "pointer" thing. You'll lose all the object inheritance capabilities though.