If you want to teach English in public schools in Taiwan, I would recommend contacting Teach Taiwan. They're very helpful throughout the recruitment and hiring process. They have a local coordinator to help you get you set up and handle issues (banking, work and residence permit, etc).
Cons
Overall, I can't think of any major downsides to finding a public school teaching job in Taiwan through Teach Taiwan.
You get a housing and flight stipend. Kids are normally great. Working at a school is more consistent than a cram school.
Cons
agents often have little to no experience. If you get a good agent your experience will be great. Otherwise, it might not be. Lots of bureaucracy. The salary is lower than a private school.