I recommend www.iss.edu for placement and the autumn is the perfect time to apply. I.S.S. (International School Services) has been around for MANY years, well established, with a golden reputation. They do their recruitment conventions early in the spring. (Several readers of my blog recommend Search Associates as a placement agency.)
They do charge a fee BUT most schools will reimburse you for that as part of the hiring process. I.S.S. represent schools all over the world. If you have limited experience (under 3 years) you will be limited in the schools you can apply to. There will still be many, but likely in developing nations or the third world. (That's not a bad thing at all!) With 3 or more years, you can go almost everywhere. Keep in mind too I.S.S. is not the only "game in town." The military has it's own school system, and their are other placement agencies. I did my own research and found I.S.S. was a better fit for me, but you might want to explore other options.
To teach in a few rare places, like Saudi Arabia, you might have to be a teaching couple. Some schools do not allow single teachers, while others do, so situations may vary. Their pay though is astronomical, usually well over $100,000 per teacher, and most of it is tax free. Some schools are aligned with the airline system and teachers fly free all over the world. I have heard of teachers working there for a couple of years, coming back to the USA, and buying a home with cash. The downside is that you have very little freedom of movement, and if you try to get or make alcohol, they'll chop off your foot. (Happened to a teacher in 1990 who built a still in his apartment, went to jail, got a warning, did it again, lost his foot.)
Being a teaching couple can present different issues because you have to find a school that needs BOTH positions, so it can be a bit more challenging. On the flip side, if accepted and you have children, they often can attend free. Some schools will do this for single parents as well, but you'll have to ask.
Consider too the political climate of a country. Most schools through I.S.S. are accredited back to the US system. On campus, it will feel like an "American School." But if, for example, you are an LGBT person, you might not want to go to a country where LGBT people are imprisoned or worse. Some countries forbid religious expression, or only allow the local faiths to be expressed. These are questions to ask during your interview. Be candid about your concerns, schools will be candid in their replies. That said, expats (Teachers from the USA teaching abroad) are often held to a different and more forgiving standard. While locals may feel some persecution for their differences, expats may not. These are rare issues, but something to think about.
What art teacher wouldn't LOVE to teach in Paris or Rome?!? "Primo" spots may pay poorly because everyone wants to teach there (Paris, London, Madrid, Rome, Milan, Tokyo, etc...) But if you are willing to go for an adventure, like I did, the pay is generally good, but if it "looks low," dig a little deeper. They usually have lots of little bonuses. Schutz American School of Egypt pays about $20K per year or so, BUT when I was there from 1991-1993, I got free room and board, use of a car, maid, and got to buy flights and cruises at Egyptian resident rates (Not tourist rates). You can live like like a Pharaoh on that kind of money, and it's all NOT TAXED!
I use to take a tram downtown, have dinner, see a move, sip a cappuccino on the Mediterranean sea, for less than $10. The average Egyptian makes about $14K per year and have to pay rent, food, car, and more out of that... You get 20K free and clear... When I came back to the US after 2 years, I had enough cash to buy a car outright.
As a teacher I was the resident expert. I was able to design classes I thought would be connected to curriculum, and engage the kids. I taught a semester of Kite-Making, Puppetry, and the regular k-12 art classes. It was a JOY! When I was at the school, I was in charge of a graduating class of just 7 students! Though many classes were a bit larger, it was nothing like the sardine-packed classes back in the states.
Obviously as an artist, I was able to paint and draw at the Pyramids, sketch along the Nile River, visit Jerusalem, Athens, and other locations that were nearby. I even had my work bought by ambassadors and art collectors of the area.
If you can do it, I recommend you give teaching abroad a try.