Let's assume that we also get to redesign our high school mathematics teachers as well, in the sense that we can assume that they know and can teach whatever material we choose to cover.

This is similar to these MO questions, although the first is more elementary school level, and the other two undergraduate.

I'm inclined to agree with much of Lockhart's Lament, on the rote learning vs. creative mathematics, although I would still worry about missing out on basic skills. On another issue, I know of at least two people questioning calculus as being the "goal" of high school mathematics, and advocating other subjects (Arthur Benjamin and less directly, Igor Rivin). Are there competing opinions?