I did this when I was your age. Great experience. To answer your specific questions, (1) I showed up first, and then looked for a job; (2) I was in Europe about 5 years; (3) really depends on who you are, what skills you have, and what you’re looking for (although I don’t trust Argentina because their Spanish is incomprehensible and they keep making imperialist attempts to conquer the Falklands).
I would recommend somewhere outside of the English-speaking world, and not only if you’re interested in a language: it’s when you cross a language barrier that you really find yourself confronted with cultural differences on a constant basis. Of course, it also makes everything much harder, but if you want easy, stay home.
Assuming you’re planning actually to move somewhere, and not just blow your savings for a few months, you have two main issues: (1) getting a visa and (2) getting a job. I have useful passports, so I have nothing to say on the visa front, but unless you have a highly desireable skill, or important contacts, this can be very difficult. I don’t know what your skills are, but many skills don’t cross national boundaries so easily (computer stuff, which I did, is one major exception).
Things to keep in mind… German-speaking and Scandanavian Europe (and Holland) all have an amazing level of English, so there are sometimes good jobs there for people who only have English (I showed up in Germany without a word of the language and proceeded to work for years in one of the top companies, and I know plenty of people who have done similar things) - and they do appreciate it if you’re trying to learn their language, particularly if you don’t have to. Also, my experience has been that pretty much anywhere you go, if you’re an educated native-English-speaker, English teaching jobs are often available once you’re on the spot (assuming you have a work visa) - although teaching English pays pretty crap in Europe (it can pay quite well in the far East, I’m told).
To give more specific advice, I’d have to know more about your particualr scenario, but do keep in mind that culture shock can be a very real, and very powerful thing. It’s all exciting and different for a few months, but for me, there was a sort of hammer-blow around the six month mark. Things got really hard. If I had known how hard it would be, I might not have gone in the first place, but in hindsight, all very worthwhile.