Hey there! It's so cool you're interested in this.
If you're interested in languages, or if you know a second language even slightly, there are lots of places (especially in Europe) that will give you cheap (or possibly free) accommodation in return for consistent language lessons.
On a more personal level, I really recommend
Rustic Pathways. I did a one-week volunteer program with them for schoolies, and I heard a lot about their other trips. They do adventure trips, volunteer trips, sightseeing, tourism, adrenaline, the lot.
On a more vague level, I have a friend on Facebook who just travelled through India, Nepal, Thailand, Vietnam, and possibly a few other places, for three months non-stop completely solo. She seems to have organised most of it herself with a travel agent and then just went for it, and apparently it really wasn't that expensive given the location of where she went. She backpacked, went sight-seeing, did a few little tours with organised groups throughout, etc...
Or, if you're looking for something well-known and probably on the pricier side, Contiki does a 6 week Winter Wonderland through Europe. You might find it useful to look at their tours, even if not to actually use them, but to get an idea of how they organise their trips geographically so you can fit in as much as possible, wherever you go
Also remember that you can do an exchange at most Universities. I'm on exchange in Italy now and it's wonderful. So as much as using this time wisely is such a great initiative, also know that this is just the beginning of your travels