Having been to 14 countries during my trip (though over 83 days), my favorites and suggestion would:
Berlin
Rome
Amsterdam
I really liked Krakow, Barcelona, Florence, Prague, Munich, and London as well.
But for 14 days I would do: Rome 5 days, Berlin 5 days, Amsterdam 4 days.
Also,
www.ryanair.com www.easyjet.com
Screw the train, this can be cheaper if you book in advance. You could probably fit in a 4th city if you wanted to, but I Berlin and Rome have a LOT to offer. Amsterdam is more than just weed, it is a really cool city, has some cool museums and other stuff.
Also, take free tours. In the three cities I mentioned there is a free tour offered,
http://www.neweuropetours.eu/ . You just pay a tip at the end (you don't have to, but don't be a dick). There around 2-3 hours and are great for meeting people, getting to know the city/where to hang out, a lot of history.
Don't go to Paris. It
can be nice and there are a LOT of stuff to see, cool stuff, but like I said, I'd suggest a lot of cities over it.