What you seek is called Modulation.
The II-V-I your jazz friend told you about is pretty easy. It's really common in jazz. First you need to establish that you are in E minor, so you'll need to play something like II - V - I in E minor and then II - V - I in A minor. That's that. As you can see here for the song Nostalgia in Times Square by Charles Mingus. On bars 9,10,11 he changes keys just by playing the II - V - I of the new key (on the first two II- V -I he is omitting the I).
But if you want to make it sound better you could try two ways:
- Find a common chord. The chords of E minor are:
I:Emin
II:F#dim
III:Gmaj
IV:Amin
V:Bmin
VI:Cmaj
VII:Dmaj
The chords of A minor are:
I:Amin
II:Bdim
III:Cmaj
IV:Dmin
V:Emin
VI:Fmaj
VIIGmaj
As you can see, the IV of E minor is a A minor, which the I of A minor.
So what you could do is:
Eminor:I- IV-V - I (so that you establish that you are in E minor) and then I - IV (here, as I said, the IV,which is A minor, can be used as the I of theA minor scale).
-- The italic progressions are on the A minor scale
So we have I - IV - V - I - IV (I) IV V I
And the chords would be: Em, Am, Bm, Em, Am, Dm, Em, Am.
And you could do the same to go back to Eminor scale or whichever other scale has a common chord with the scale you currently play.
- You could change the scales chromatically. Which means you play a chord and then you change some notes chromatically and it becomes a different chord. For instance:
The VII of E minor scale is Dmaj and the IV of A minor scale is Dmin. So you could play:
I - IV- V -I - IV - VII and then on the VII lower the third (F#) chromatically and you get F natural. And the chord would be Dminor, which is the IV of A minor. So you'll have:
I - IV- V -I - IV - VII - IV - V - I
The second one might not sound really good at you at first. The chords I chose might not be the best example, but both of the above methods I mentioned are acceptable
Τhere is also another way where you find harmonic chords. Like when you play G# major you could say it's Ab major and then continue to play like you are on Ab major. But I'm not 100% how to explain that, I'll just confuse you further.
should I go "up" to new key (i.e. bass of the first theme is E3 and I move it up to A4) or "down" (bass from E3 to A2)? Which one feels more like an "arriving" to the new theme?
Τhat would depend on the player (in my opinion). I would say that if the melody becomes more intense, you should go up and if the melody slows down and becomes more dramatic, you should go down -- but that is just my opinion. Some other musician could say the exact opposite and it would be acceptable.