Does anyone know of any good reference about the representation theory of two-step nilpotent Lie algebras, like whether their irreducible representations can be classified?
Yes, all irreducible representations of a nilpotent Lie algebra are $1$-dimensional by Lie's theorem (over the complex numbers). For representation theory in general of nilpotent Lie algebras see also our article concerning Faithful Lie algebra representations of nilpotent Lie algebras. For $2$-step nilpotent Lie algebras of dimension $n$ we can prove that there exists a faithful Lie algebra representation of dimension $n$. This is not true in general for higher-step nilpotent Lie algebras.