Gap year

If you work the summer to earn enough for just a short trip, you would be taking an entire year off of school to end up working or staying at home for 10 months of it. I would work the summer, go to school (maybe work part time?), save up some money, and then look at doing some travel on your summer breaks, or wait and plan a longer trip after you graduate. Travel will always be there, and school offers a lot of breaks travel during. On the other hand the universities only admit once or twice a year, and a good education is an important foundation for the rest of your life.

If you don't have money, what's wrong with working? You can work at home, if there are jobs in Greece, or you can work on the road. As a teenager, you can pretty much guarantee that every job will be pretty crap wherever you are, so the only benefit of staying at home is to leech off your parents - and the whole point of travel is to be master of your own destiny, right?

Off the top of my head, I read last summer that tourist spots in Portugal were having problems recruiting local workers. (The season is five months, but you have to work six months to qualify for social security afterwards, so anyone who doesn't have a job and already getting social security has a disincentive to work.) After the summer tourist season, it's time to go pick fruit. The most famous option is grapes in France, but there are hundreds of others, and in fact there's a website dedicated to this. Google is your friend.

When that's over, you're into the colder weather and will probably want to head for the sun - Cyprus, Egypt, Morocco or Southern Spain (Canaries, maybe?). Every other itinerant in Europe will be heading the same way, so things get harder at that point. Or you could think about the ski season, Christmas rush, and so on in wealthier countries. Then it's planting time on the farms, and so on ad infinitum.

And if you can convince people you're a native speaker of English then you can always find teaching opportunities in some countries.

University will still be there when you come back, and so will the pressure to take the next steps in your career. I meet so many people who have reached middle age without ever finding the time to make that big trip. If you're going to put it off until after some other milestone, there will always be another milestone. Either it's a priority and you do it now, or you forget about it.