Figuring out the Lagrange interpolation polynomial was a pretty awesome moment for me as a high school nerd.

I was amazed a while later that you can simulate a Turing machine with just two counters, but that takes a bit of technical stuff to explain what a Turing machine is.

$x+1/x\ge 2$ if $x > 0$. Proof: $(\sqrt x-\sqrt{1/x})^2$ must be >=0, so expanding, $(x + {1\over x} - 2) \ge 0$. Not very deep, but kind of an aha moment in seeing reasoning appear from nowhere and immediately look obvious, getting rid of a calculus problem.

Proof of the triangle inequality in R**n, using Schwarz's inequality. Again, maybe the proof isn't beautiful in itself, but it was eye-opening in connecting geometry to analysis.