I would say that a young person could make it through a decent amount of Central/South America w/ money saved from a weekend/night job. I knew someone who backpacked most of central America, in about a month and half, and he did it with money he'd saved himself. I would have done it as well, but whenever I head down there it's for visiting family and not really for vacationing.
The countries in the EU cost more money for us Americans, of course, so you might not be able to get as far there nowadays.
I backpacked through Spain on the cheap (staying at hostels, etc.), but I did have to dip into my parent's money a little; the dollar was doing terribly against the Euro the year I went.
I'd encourage anyone to travel/study abroad. As cliche as it sounds, it really can change your worldview. One of my close friends in college was a borne-and-bred southerner with some very entrenched beliefs. He'd never been out of the country until after we graduated, when he backpacked through all of Europe. After that experience and a summer in Americorps, he has reevaluated his stance on many things (including gay rights, hooray!) and I believe he is a much better person for it.