I am trying to find all topologies on the set $X = \{1,2,3\}$ that are $T_0$, $T_1$ and then all that are $T_3$. I believe I am good with which ones are $T_1$.
A $T_3$ or regular space is for every $x \in X$ and for every closed set $F \subseteq X$ not containing $x$ there exists open sets $U$ and $V$ which separate $\{x\}$ and $F$. I'm not sure how to apply this to my question.
There is of course but one $T_1$ topology on a finite set: the discrete topology. The discrete topology is also regular. Another obviously regular topology is the indiscrete (trivial) topology: there are no pairs of closed sets and points to separate at all, except $F= \emptyset$ and any $x$ (where we are forced to use $U = X$).
A space that is both regular and $T_0$ is $T_1$: Let $x \neq y$. Then say we have (by $T_0$) an open $O$ with $x \in O$, $y \notin O$ ("half of $T_1$-ness). Then $x \notin X\setminus O$, and the latter set is closed, so (by regularity) we have $x \in U$, $X \setminus O \subseteq V$ and $U \cap V = \emptyset$ for some open $U,V$ open. Then $y \in V$ and $x \notin V$ showing the other "half" of $T_1$-ness.
So if we are looking for non-discrete regular topologies, we need to look for topologies that are not $T_0$.
Here is a list of all (non-homeomorphic) topologies on a three point set $\{a,b,c\}$:
This is up-to homeomorphism, through permutations of the points we get more topologies (I explain that here) but no essential new examples.