You could use multiple USB microphones and do the mixing in software. I prefer using an audio interface (e.g. UMC204HD) because it is simpler to set up and adjust levels, and because it lets you use any widely available microphone, or plug in an instrument (e.g.: electric guitar, electric piano). You can plug your headphones into the audio interface and adjust the relative level of your own sound and what is coming from the computer (e.g.: your teacher). sweetwater.com has the UMC204HD and the UMC404HD on sale right now.

I do not have any experience with pick-up mics.