What does 这样才不白来世上一回 mean?
我什么都想试试，这样才不白来世上一回。
  
  I want to experience/try many things, only then will I feel like my life hasn't been wasted.


That is the general meaning. As for the bolded part:


這樣才 - only like/by doing this, referring to 我什麽都想試試
白X - a waste of effort to X
來世上 - be born into the world
一回 - one time


來世上一回 is a figure of speech culturally referring to the possibility of reincarnation (being born multiple times). The idea is that coming into this world is a journey that takes effort, and while being here this person wants to experience as many things as possible.