My honest advice would be to shorten your itinerary. In my opinion, it is much better to see a few places in comfort and thoroughly, meeting the locals and enjoying yourself, than to spend all your time driving frantically around the continent and ticking locations off a long list! I spent nearly six months travelling in a campervan around Europe, and didn't see the number of places you and your mates are planning to see -- there's no need to aim so high.
It's natural to want to see a whole slew of places, but really if you just have three weeks, three cities is ample.
I advise sticking to the Prague, Berlin, and Amsterdam section and only that section! Spending four or five days in each of those cities would be a fantastic experience, and gives you time to get to and from each one, plus side trips. If you are on a really low budget, bring a tent or two and you can camp. Prague, Berlin and Amsterdam have great, cheap, clean, and lively campsites near to the city centre (I like Camping Zeeburg for Amsterdam and Sunny Camp for Prague, but I stayed with friends in Berlin, so can't recommend one personally there.)
Plus it's a relatively simple drive between those cities, in terms of them being fairly close geographically, and with a decent road network and signage. I didn't have a map and made it, so how hard can it be? The other cities you mention have their charms too but might be a little bit uptight (i.e. Paris) or not have enough in them to sustain your interest for more than a day (i.e. Verona, Milan). Amsterdam, Berlin and Prague don't just have oodles of sights and lots of interesting culture, they also have a thriving night life, so you would have to work really hard at being bored, even if you stayed there a month. :)
So yeah, to summarise: Keep it simple and you'll enjoy it more. I hope you have a great time.