How to Travel from ChingShui to Tainan and Lu-Gong

I suggest you learn how to use google maps
https://www.google.com.tw/maps/
First it can help you find the names of the cities (English and Chinese, so no confusion)
Next it can help set up your trip (car, bus, train, on foot + estimated time and budget)
Finally, you can learn about the local geography
Another website is http://herewego.com/ (I use it for bicycle trips)
Good luck!