You have a good start - you need to deal more explicitly with the elements for the critical step, however. In particular, you can write $H = \{g^{n[G:H]} : n\in\mathbb Z\}$ and $K=\{g^{n[G:K]} : n\in\mathbb Z\}$. A reasonable thing to do is to ask which elements are in common between these sets. Note that if you already have a strong handle on sets of the form $n\mathbb Z$, you may be able to very quickly get this result by using that - but if not, you can also do things in a more group-theoretic fashion.
First, observe that $g^{\operatorname{lcm}([G:H],[G:K])}$ is in $H\cap K$ where $\operatorname{lcm}$ is the least common multiple, since the exponent is a multiple of both $[G:H]$ and $[G:K]$ by definition. Let $R=\langle g^{\operatorname{lcm}([G:H],[G:K])}\rangle$ be the subgroup generated by this element. First of all, note that the exponent divides $|G|$ since both $[G:H]$ and $[G:K]$ do, thus the order is just $\frac{|G|}{\operatorname{lcm}([G:H],[G:K])}$. Then you can use some number theory to observe that if $a$ and $b$ divide $n$, then
$$\operatorname{lcm}\left(\frac{n}a,\frac{n}b\right)=\frac{n}{\gcd(a,b)}$$
which essentially follows by noting that reciprocation reverses divisibility.
Applying this with $n=|G|$ and $a=|H|$ and $b=|K|$ and using the formulas you gave for the indices of the subgroups gives that $|R| = \gcd(|H|,|K|)$. Then, we are almost done since we know that $R \subseteq H\cap K$. The only remaining step is to note that $R$ is actually all of it - but we know that $|H\cap K|$ divides both $|H|$ and $|K|$, so the intersection cannot have more than $\gcd(|H|,|K|)$ elements, so must be exactly $R$.
