I know there are several variants of proofs for the Prime Number Theorem.
Which one is the easiest one to study and then re-teach?
By easiest, I mean those that assume minimal knowledge beyond secondary school mathematics.
For example, most school leavers having done maths will have calculus, and could stretch to understand concepts like asymptotic equivalence and integration in the complex plane, but won't have concepts like group theory.
You might check out
D. Zagier, Newman's Short Proof of the Prime Number Theorem,
which appears in
The American Mathematical Monthly, Vol. 104, No. 8 (Oct., 1997), pp. 705-708
and is available online at
http://www.jstor.org/stable/2975232.
This is the easiest proof of which I am aware, but its challenging nevertheless. But an undergrad with a solid background in analysis should be able to hack it.