How important is teaching experience?
Even if the PhD research does not go as planned, I would strongly suggest you to get some teaching experience for a number of reasons:


"One learns best when one teaches" as a translation of the German "Am besten lernt man, wenn man lehrt". Teaching gives you a deeper understanding of many topics, it helped me a lot.
Character building: you as a person will benefit as well from teaching. Speaking publicly is not an issue for me since I taught during my PhD studies, this had a really positive effect on my work outside academia as well.