The thread is already going in GD, can you take a minute to post what you suggest?
I don't know if your main goal is to live it up or educate yourself.
If you're looking for a blast - Amsterdam
If you need some historical experiences - France and Italy.
And don't forget to route your trips through Lichtenstein and Andorra, just to say you've been there.
Everyone's given you pretty good advice so far. Just read up on the places you decide to visit (5 weeks is NOT a lot of time).