Hiya Duffle
First off, I was terrified about taking my first step and leaving home. I travelled to Greece alone and was in such a state at Glasgow Airport. I damn nearly turned around and came back again because the whole thing was a massive culture shock, but I stuck at it and am so happy that I did. So I'd recommend that you allow yourself to feel a bit scared and recognise that it's a good thing! I was twice the person I was when I finished my year there; it really was an education in every way.
The first place I looked for employment was Paris. I quickly learnt that Parisian schools specialise in Business English and generally don't recruit first-time teachers. This is largely similar to Germany and more prosperous European countries where English is part of the primary curriculum.
You are more likely to find employment as a newbie teacher in countries like Greece, Turkey, Thailand, possibly Italy and Japan, where the demand for native English speakers far outweighs the requirement for experience, specialised qualifications, etc. It may not be where you necessarily want to go, but do your first year or two there, and then with some experience under your belt you can pretty much pick and choose your next destination. Greece was never my first choice but I had such an enriching year there; remember that your new home is very much what you make it!
You should look in The Times every Tuesday; they advertise jobs and have a separate category for TEFL.
I did two years of general English and one year of Business English. Business English is pretty intense and you may find yourself 'on call' to teach as early as eight in the morning, and as late as nine at night. If you teach in a general English academy (kids and possibly adults) you have a greater likelihood of block hours in the evening, for instance 4-10, which leaves you much of the day free for sightseeing or whatever else you fancy.
The pay is generally lousy (unless you go to Japan, apparently) but some schools have their own flats for teachers, and elsewhere the cost of living can be a lot lower than the UK. And you're not doing it for the money anyway, right?!
You don't necessarily need to speak the language of the country you're going to, but obviously it helps in your own day-to-day quality of life.
Hope this has been helpful. It is scary but ultimately so much fun, and so life-changing. I can't recommend it highly enough.