If $a^2+b^2=c^2$ then $a$ or $b$ is even. I am having trouble proving this directly. I know that it is easy to prove by contradiction by assuming both $a$ and $b$ to be odd, but how should I start to try to prove this? This is a homework problem and I am just looking for help getting started, not the full answer.

It's a bit of a fine line between proving directly and by contradiction. Try breaking it into cases, such as
Suppose $c^2$ is even/odd and WLOG $b$ is odd, then show that $a$ must be even.
Since the conclusion isn't a contradiction, but rather that $a$ is even, this is a direct proof. You should end with something like $a = 2*n$ where $n$ is an integer (definition of even).
It might help if you have a theorem that says that $a$ and $a^2$ have the same parity. Maybe worth a lemma.
