Getting a job teaching English in Taiwan usually involves going there to find work. A few schools may post on Dave's ESL cafe, but the majority of jobs in Taiwan are advertised on Taiwan based sites. Some of the big chain schools like Hess - the biggest, advertise abroad.
And you can do it that way if you want. Although many of the jobs that hire from abroad have some things in common like: lower pay, off the beaten path jobs, etc.
So anyways here's how I did it.
I got a tourist visa from the nearby Taiwanese embassy in my country, but goods news now you don't need that. If you are an American you can get 3 months free of the plane.
I tried to get a 2 month multiple entry tourist visa which is the recommended way, but they didn't give me that. They gave me a 1 month multiple entry visa which was really the same back then if I just showed up.
In retrospect I would of done it differently had I known that they would only give me one month. If you are from the USA you can get a free one month 3 free months for a landing visa just by showing up. I would have done that.
So I got a ticket and flew to Taiwan with about $2000. I had a couple of college friends in Koahsiung who I visited, but I didn't want to be in Koahsiung, so I went to Taichung. I stayed in hotels and did some subbing to make some money teaching English before I ever landed a job that got me a visa.
Starting off as a substitute teacher is also a good way to get your feet wet. I did that for awhile and it took me about 3 months before I found a job that would give me a visa. I probably could of found something sooner if I wasn't too picky.
Since my visa was for only one month every month I had to take a visa run to Hong Kong. I flew there three times, before I got an ARC and resident visa from my job.
The tickets were pretty cheap and maybe cost around $150 then. At one of the schools that I did some subbing at I made a Taiwanese friend who helped me find some cheap short term accommodation.
Aside from the subbing that I was doing I found my first job by going door to door in my neighborhood. I went to all the buxibans in my neighborhood and found my job which was pretty easy going. The good thing about this is that you get to see the school in person. That's how I got my first job. So that's another tip on how to get a good job.
I'd recommend just going to the country and looking (but not if you are a chicken sh**, hehe), but it depends on the country.
Learn more about teaching English in Taiwan.