To answer your question incidents about a song written in both C minor and A minor. I can't think of a song right now that modulates like this but it is a common key change. It happens very natural too. Here's a way to go from A minor to C minor. In A minor the dominant 7 chord is G, and G is a fifth above C. In the C harmonic minor scale G is the dominant. So you just play a harmonic minor cadance going from G7 to C minor. So you can easily go Aminor - G7 - Cminor. To go from C minor to A minor, you can just play back in the circle of fiths. This is done by going clockwise from the key of C minor to A minor in the circle of fiths. The chord progression would be, Cm - Gm - Dm - Am.

Beginner-

Listen to your teacher he's got the right idea. Your goal although should be to know every note on the fretbaord that is in the key you what to improvise in. So if I said play the G major scale starting with the 15th fret on the low E string, you should know it all the way up and back down without thinking about it. Learn your fretboard, it will help.