I couldn't find any reference with the representation theory of this specific case. I found some general stuff but never explicit computations or realizations.
The only thing I found on $SU(2, 1)$ is a little bit about its structure theory on Helgason's book (Differential geometry, Lie Groups and Symmetric Spaces).
And so I ask you, do any of you know if there is something like this in the literature (I phrase this question as a reference request because I think it would be near impossible to give an explicit answer in here)? Actually, at this point, anything on this group would be great, but I do need to know whatever is known about its representation theory.
Bonus question: 1) Refenreces on the relation of the rep. theory of $SU(2, 1)$ and $SL(3)$ 2) The relation to the rep. theory of it's Lie algebra. 3) References on actions of $SU(2, 1)$.
The answer depends on the meaning of "representation" here. In case you mean "unitary representations" by it, a possible reference is The Structure and Unitary Representations of $SU(2,1)$, where also the finite-dimensional representations of the Lie algebra $\mathfrak{sl}_3(\mathbb{C})$ are classified, and all of your bonus question are answered. A special reference here is $[10]$, a complete PhD-thesis on this topic for $SU(2,1)$ and $S(U(1)\times U(1,1))$. There are many other references as well.