Best places? Very hard question.
I think it comes down to preference a lot. I personally really like Nottingham. Most people hate it.
Generally people agree that Berlin is awesome and if it wasn't full of Germans, I'd totally agree. I've spent most of my life in that city and it's never gotten boring. I just didn't like all the Germans and wanted a change. For tourism or living there for a couple of years it's brilliant and it also is the town that gets the most tourists in the world and large enough to still be full of non-commercialised awesome.
The coast of Poland is awesome, as well as the Frisian coast. Greece has loads of really old stuff that I really enjoyed and nice vegetation. Austria is pretty in the winter, generally, if you go into the mountains and stuff. Most Southern European countries have areas that are pretty much tailored toward American tourists, so if that is your sort of thing, just go with a travel agency. For countries where you don't speak the language, everything else can be quite scary if you don't pay a guide, especially if you're not used to travelling, but you'll get by fine with English in most places; just rural areas and some parts of Eastern Europe are a bit less anglicised. Just take a look at what sounds appealing to you.
For living, it's all different if you're European. I get full health care coverage in most EU countries, for example, but you wouldn't. You'd also need a visa where I don't and other stuff. Just pick out a country you like in some way (and if it's only their food) and visit the embassy website, they always have some good pointers.
It's really all down to preference and how much money you have, what you plan to do etc.
I know that quite a few people take a gap year and just travel; one of the brilliant things about Europe is that there are very diverse cultures all on one fairly small continent, so you could do one of those work and travel things and live right out of your pocket, or go fully gipsy and just play the guitar badly in public places, you can live surprisingly well off that if you keep to the bigger cities. I know. :P
You also might want to check out this
, I've known a girl who did it in Spain and is now fluent in Spanish. I'm sort of considering doing it myself at some point, after my BSc maybe.
There are various international volunteering programmes, I don't really know any others.
And I can't stretch this enough, I don't know what your preferences are. I mean, surely you know if you prefer cities or the countryside, flat places or mountains, hot or cold weather? You'll probably have fun anywhere simply because the places will be new and exciting to you even when they're really dull to anyone from the area.
I think you should maybe wait till you've finished High School or are legal or whatever, anyway. I mean, I've only started travelling on my own without anyone to organise stuff for me a good year ago and it probably would've been proper chaotic before that. This
is also a good place to look at if you're not afraid of the whole thing, we used to be hosts for a while and it's a great way to meet interesting people and get an insider's perspective on places you visit. Or make friends on Allakhazam! I've visited a few people who post here and so far they've all been really nice IRL and there's actually some who are more awesome IRL than on the forums.
As for languages and stuff, Skype is the best thing since airplanes were invented. I think I mentioned that I am working toward Advanced Level Dutch without even having done the basics before (apart from the very
basics) over Skype with two nice people who post here for teachers? It's the fastest I've learned a language outside of the country where it's spoken. Ever.
Also, if you get to planning a trip, feel free to ask me how to optimise, for anything within Europe I pretty much know how to do it or at least where to look.
TL;DR: Best places to visit can't be defined without knowing the person planning a trip, alternative methods of travel are awesome, Skype is a great language learning tool, and contact me for more wall of text!