Question: Homework question, so just a pointer would be nice, for starters.

I'm trying to prove $2 \mid 5^{2n} - 3^{2n}$ by induction.

I use $n=0$ as the base step, and assume $5^{2n} - 3^{2n} = 2k$ as my inductive hypothesis. Then for the inductive step I do
$$5^{2n+2} - 3^{2n+2} = 25\*5^{2n}-9\*3^{2n}$$
and then I have no clue what to do, as I see no way to make the expression nicer.

Answer: 
As explained by Daniel in the comments, writing 25 as 16 + 9 allows us to use the inductive hypothesis.
$$5^{2n+2}−3^{2n+2}=$$
$$=25\cdot 5^{2n}−9\cdot 3^{2n}=$$
$$=(16 + 9)\cdot 5^{2n}−9\cdot 3^{2n}=$$
$$=16\cdot 5^{2n}+9(5^{2n}-3^{2n})=$$
$$=16\cdot 5^{2n}+9\cdot 2k=$$
$$=2\cdot (8\cdot 5^{2n}+9k)$$
$$\Box$$
Here is an alternative, simpler way:

* Prove that if $x$ and $y$ are odd then $x-y$ is even.
* Prove that if $x$ and $y$ are odd then $xy$ is odd.
* Prove that if $x$ is odd then $x^n$ is odd. Use induction here.

Now note that both $5$ and $3$ are odd.