What year? What experience do you have? What math have you done? What are you interested in?

Number theory.

+ What is modulo arithmetic?

+ If p is prime and 1 <= a < p, why is a^(p-1)-1 a multiple of p?

+ If a prime is conguent to 3 mod 4, why is it never the sum of two squares?

+ If a prime is conguent to 1 mod 4, why is it always the sum of two squares?

Topology

+ Find examples of why path-wise connected is stronger than "not disconnected".

+ Show that in 2D if you consider parallel lines to converge at infinity, and that they all converge to the same point, then what you have is a 2-sphere

+ Show that in 2D if you consider parallel lines to converge at infinity, and that non-parallel lines converge to different points at infinity, then what you have is a Moebius strip with its edge glued to a disk. (projective plane, or RP2)

Analysis

+ Understand why the sum 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ... diverges (and what that means)

+ Understand why the sum 1/2 + 1/4 + 1/8 + 1/16 + ... converges, what that means, and what it converges to.

++ Note: most undergraduate mathematicians get this wrong.

There's a bunch of stuff, and this is all straight from the top of my head. It's not necessarily good advice, but they are a few things I found interesting when I was 12 or 13.