Traveling around Taiwan

Hi there! I'm planning a trip there in end Jul till early Aug and would like to seek advise if the period is appropriate and if the typhoon and heavy rain will occur very frequently? I've read it is the typhoon season and the weather is hot and humid. I would like to go places like sun moon late for sight seeing and would certainly not want any typhoon warning to spoil that.

Also, what is the recommended transport choice let say we are going many places from taichung to taipei , kaoshiong, sun moon lake etc.
If it's by mrt, what is the usual ticket price and is there a unlimited ride ticket?

Appreciate your help!