What does it mean to prove if $p$ then $q$? Does that mean to prove $p\rightarrow q$ is a true statement? Then since when $p$ is false, $p\rightarrow q$ is vacuously true, do I only have to prove $q$ is true when $p$ is true?

That is the gist of it, yeah.
There are, in practice, several ways to do this, and here is a short summary. A direct proof uses intermediate, already-known implications chained together like this.
$$
p\to p_1\\
p_1\to p_2\\
\vdots\\
p_n\to q
$$
A contrapositive proof is a direct proof of the statement
$$
\text{not }q\to\text{not }p
$$ and a proof by contradiction is a direct proof of the statement
$$
(p\text{ and not }q)\to \text{contradiction / absurdity}
$$
