I teach English in Taiwan now (in Kaohsiung).
There are plenty of English teaching jobs here, but also plenty of teachers, so don't be surprised if there are many other applicants for the positions you apply for.
Experience is very useful - sometimes you won't even get an interview without teaching experience. People like to live in Kaohsiung long-term and many teachers here have years of experience, ESL qualifications (CELTA etc.); some even have teaching licenses. In theory, you can get a good teaching job here with just a degree; in practice, the flood of applicants for positions means that schools are getting ever more picky about who they hire.
Bear in mind that I'm talking only about Kaohsiung, which is a popular destination for foreigners. The situation may well be easier in smaller cities.
You'll need to come here, get somewhere cheap to stay (a hostel is fine for a week or two), and get out there on the streets handing out resumes at all the language schools you can find. Interviews usually include a demo lesson, so be prepared for that. Check the web for local foreigner newsgroups for job ads (here in Kaohsiung we have the Kaohsiung Living websites and a Yahoo newsgroup with the same name).
Bear in mind also that you'll probably be asked to sign a one-year contract for any job you apply for. True, there are not usually any big incentives to finish your contract like there are in eg. Korea, and the labor market is much more fluid, but giving Taiwan at least a year should enable you to save some money and enjoy the experience of living here. The first 6 months can be pretty tough.