Yes I did it on my own at 21. I think they would be fine with a rail pass and hostels. I don't think a tour would be as much fun. You really get to know the culture walking around the cities vs. driving through on a bus.
I had great experiences all over Europe- but my favorites were Italy and Greece.
I also did tours in high school and they were fun but I was ready to do it on my own by the time I was 21. I lived off that Let's Go Europe book.
cracking up that the place I stayed out for "one night" that turned into 3 nights is still around
http://www.thepinkpalace.com/
$12 you get overnight lodging, breakfast and DINNER! (back then it was half a chicken with veggies) for $12~ I paid $10 25 years ago back then you could sleep on the roof for only $4 and get breakfast and dinner. And I'm still alive!