Teenagers just like young adults share the same fate: plenty of time at hand to travel, and the drive to do so, but in most cases they lack the funding to comfortably travel around the world.
Transportation is one major cost factor, and the well known possibility of hitch-hiking is no longer that popular and will not get you across the ocean. We suggest to combine the journey with a bit of added earnings, which might come in handy later: organize a job which will get you exactly where you want to go to. Work on a boat, contract with a logistic firm running special errands for them, accompany people who need help like elderly or handicapped people. Alternatively you can at all times do a fund-raiser so that you can afford a low-cost plane ticket.
When you get to where you want to be you next worry is food & board. Even though youth hostels and street vendors or supermarkets are a low budget option, if you really like to stay somewhere free of charge and also be fed, there are mainly 2 alternatives: One choice is that you are willing to work locally. For example farms will host you, provided you help with the work a number of hours every day. Another option is to work as an au-pair: for twenty - 30 hours every week of babysitting you receive food & board, sometimes a car, and all of that not just for free: you get paid.
If you don't want to work, a private student exchange will be your best choice to travel for free. Your host family will provide you with food & board for free. In return you later host their teenager at your home. Of all student exchange programs only a few offer a private student exchange. The concept is easy: I visit you - you visit me. Such exchange student programs even allow family hopping and multiple country visits. For example you can tour all of Europe and stay with families in Paris, London, Rome, Berlin, Munich Barcelona, or Madrid for free.