You may want to take a look at
Learn Chinese in Taiwan for further information.
Students can study Chinese Mandarin either in Taipei or Taichung. Many students will also have full-time English teaching jobs at the same time. This is quite manageable considering that the Chinese lessons are only 2 hours a day, 5 days a week. Typically you can choose either morning, afternoon or evening classes, depending on how popular the courses are throughout the year.
Keep in mind though, whilst you can teach at the same time whilst being enrolled in a Chinese classes, technically speaking you will need an ARC (work permit) to teach legally.
The new semester for studying in Taichung starts beginning of September. So first decide which location you want to be in. Do you prefer Taipei or Taichung? This is only something you can decide.
September and October are great times to arrive to Taiwan as recruiters and schools are going crazy trying to fill teacher placements.