What you seek is called [Modulation](http://en.wikipedia.org/wiki/Modulation_%28music%29).

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](http://www.realbooksite.com/jazz-sheet-music-images/Jazz-Sheet-Music-Page-326.jpg) 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:

1) 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.

2) 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.