What basic kinds of knowledge of C++ should I acquire if I'm to write a music notation software?

MuseScore is open source, you can do about anything with it. And no secret about it either, nor rude to ask this

Aside from learning a programming language, a bit of patience is at its place as well; especially in worldwide timezones and during the weekend ;-).

First things first; music notation is not simple. This is not at all meant as a discouragement, but just so you make sure you have a clear goal of what you want to achieve in the often limited time a course project gives you. It could be a great start-up of something to use later on though.

The most important thing to understand from a C++ perspective is how the concept of classes is used to translate a real-world model into a computer-useful model. How you can make objects interact and when to use inheritance to limit the amount of code to write.
It is always a useful architecture to separate the data model and what can be manipulated (in MuseScore sources "libmscore") from how it is displayed/interacted with (in MuseScore sources the "mscore" folder); especially in C++ where there isn't a default GUI framework available.

For MuseScore, that visualisation and interaction is done using the Qt libraries, as they provide us with cross-platform capabilities even outside of just the visualisation.
A more lightweight toolkit is SFML, which offers C++ graphical and audio capabilities among others and also claims to be cross platform. It is built on top of the lower level SDL library, for which an abundance of resources can be found as well.
If you're working on a specific version of windows only and don't care about cross-platform capabilities, then looking into MicroSoft's .Net platform is also an option.

In my experience, once visualisation starts, it's more a question of how well you know the specific libraries and their quircks than being excellent at C++ (although that of course doesn't hurt ;-) )

MuseScore is open source, so feel free to browse around and lend what you seem fit. Be aware that the people judging your end result might also know this, so for an assignment its probably not so wise to copy-and-paste core functionality if your own contribution to the end result is diminished by it.

Finally, (once more), the biggest factor of success is being able to clearly scope your project. Be clear in what you want to achieve, write it down. Doing so will help you in finding out which classes you'll want to represent the data you'll be working on.