I want to get to Zhongshan from Taipei and then go to Guilin. Any suggestions?

You can go to HongKong, Macao, Guangzhou, Shenzhen or Zhuhai from Taibei, just find which city is easy for you to catch a flight from Tabei. Then take the ship/boat from HK or Macao to Zhangshan, or take the bus/taxi from Shenzhen or Zhuhai to Zhongshan.

Then go back to Shenzhen to take a flight to Guilin, which is only 1 hour.

Enjoy your time and have fun.