I just installed BOINC manager on my Windows 10 computer, and I couldn't add any projects whatsoever. Later, I got a notice within the manager that it could not connect and asked me to check my connection. What should I do?

That sounds suspiciously like a firewall is getting in the way.
First, have you allowed BOINC to pass traffic through to the outside world (both out & in)?
Second , if you are in a work situation the company IT policy may block BOINC.

The first is normally easily solved,and the second not so easy....

Sounds suspiciously like the Manager could not connect to the BOINC client. Different type of communication error.

The BOINC client is the application that does the actual communications with projects. If you can't connect to projects, you can't add them. The BOINC Manager is nothing more than a GUI interface to the client.

Look for threads concerning gui_rpc_auth.cfg permissions problems. If the Manager can't connect to the client, you are dead in the water.