Which is considered more valuable for faculty job applications, teaching experience or participation in conferences? As a PhD student from top university, I have the opportunity of having a minor called "teaching minor". For me, I am personally mostly interested in mathematical application into material science while I am very interested in getting tenured faculty positions. 

Actually, I am more interested in going to a lot of conference and trying to advertise or publicize the result of my research and I am worried that committed teaching during the teaching minor program would render me less experience in conference or slows down my research progress.... 

Any suggestion on my situation? Shall I pursue the teaching minor, or minor in mathematics as it is my research and personal interest? Or shall I do exhaust myself trying to do both?
Where do you want to work? Research universities want people who do research, publish, go to conferences,  and can tolerably teach, in that order.

Small liberal arts colleges want people who can teach and who do enough research to keep up with the field. 

These are rough generalizations. There's no right answer here.