What are the transitive groups of degree $4$? How can I find all of the transitive groups of degree $4$ (i.e. the subgroups $H$ of $S_4$, such that for every $1 \leq i, j \leq 4$ there is $\sigma \in H$, such that $\sigma(i) = j$)? I know that one way of doing this is by brute force, but is there a more clever approach? Thanks in advance!

We know subgroups of $S_4$ come in only a few different orders: $1,2,3,4,6,8,12,$ and $24$.
If we use the orbit stabilizer theorem, we have that $|H| = |\operatorname{Orb}_H(x)| \cdot |\operatorname{Stab}_H(x)|$; this limits us to only four possible subgroup orders, as $|\operatorname{Orb}_H(x)|$ can only be one thing.
Among those four orders, we can find $5$ non-isomorphic groups fulfilling the criteria. I don't know off-hand how many copies of such groups $S_4$ has offhand, but it shouldn't be too hard to pin that down.
Note that just because two subgroups of $S_4$ may be isomorphic, it doesn't mean that they all (do or don't) act transitively on $\{1,2,3,4\}$. For example, I can only think of one particular Klein four-group in $S_4$ that does, although there are several isomorphic subgroups in $S_4$.
