Hi Roxie,
My younger brother and his girlfriend travelled around Europe in the mid-1980s. They caught the train to Dover then crossed to Calais as foot passengers. From there they just travelled in whichever direction they found themselves going and happenstance took them.
The two left with a small tent, two rucksacks and about £100 in cash. My mother and I thought we'd never see them again, but they eventually arrived home after almost three months having visited 13 countries. We often received post cards from different places they'd been. They arrived home with a whole load of stories of places they'd visited and people they'd met. They travelled around by hitch-hiking and many people took them home, fed them, did their washing, let them stay the night, gave them breakfast then saw them on their way the next day with a packed lunch. They often asked bakeries and other shops if they had any food they couldn't sell and they often got food that way.
Hitch-hiking sounded dangerous even then and I wouldn't recommend it now but that's what they did. Asking for food is also the sort of thing you can get away with when you are younger. I know it was a few years ago but hopefully things haven't changed too much. I would say don't plan a definite route. My wife and I often enjoy one-week City Breaks and after visiting the usual hightlights of the guide books we like to just walk and get lost. It's amazing what you discover when you do that. I would suggest you just travel as luck takes you and you'll discover so much and have a great experience you will remember for the rest of your life.
Good luck and have fun!