I believe you need a visa before coming to the country. Then, when you've found a teaching job, you can easily get a residence permit. Your employer will organize that and renew it as well.
As for getting a teaching job, there are some big language institutes where you can try (they have mixed classes, though, which you may not like), or you can go to private schools, especially those with an English section (they usually pay better) - there are lots. Having some teaching experience is certainly helpful but not strictly necessary if you are talented.
