Do any high schools teach public speaking?

Absolutely.  In Texas it is also a state competitive  academic category.  They teach it school, students compete at the school district level, then between districts, followed by regions, and finally at state competitions.

afaik, not in a serious manner.

My children's high school offers a public speaking course.  It's not required - it's an elective they can choose to take.

Some do. In others, the concept of public speaking is folded into many other classes where students make oral presentations. 

in Australia we have debates where teams debate Infront of the school