What Taiwan City Should I Live In?
There are English teaching jobs in most places in Taiwan. In big cities, there are more people which equals more students which equals more schools and jobs. Urban areas from North to South have their own foreign community and job market. Salaries may be higher, depending on where you work and your experiences, but the cost of living is also higher.
If you are looking for a quieter and less urban Taiwanese environment, areas like Tai Tung, Ping Tung, Hua Lian, or Chia Yi may be good choices. These areas are small and there may not be many full time positions available. However, being a foreign teacher in these areas is a novelty and many schools would be happy to have a good one. You will certainly be competing with fewer people for jobs. Pay may also be lower, but so is the cost of living.
Also to consider is the foreign community. If you are looking for more foreign teachers to hang out and be social with, then finding a job in an urban environment may be more suitable. When you move to smaller cities and towns, there may be a very small, if any, foreign community. This has its advantages as you will have plenty of chances to work on your Chinese and learn a lot about Taiwanese culture.
When finding a place to live and work in Taiwan, it all comes down to personal preference.