Hi, I may take a holiday and sight-seeing in Taipei, Taichung from mid April to mid May/2014.

Yes, there is a rain period.
The plum rain will happen in May, but please do not worry, it will not rain heavily and rain all the days.

If time is possible, both Qing Jing Farm and Yang Min mountain are both suggested. Besides, the Maokong Cable Car, the Shilin Night Market, Hsimenting are also suggested.
If you want to aviod the train, it is suggested to keep attention on the weather forecast and arrange the trip according to the weather to different places.