I know of two clean approaches to classifying the unitary irreps of the $ax+b$ group. The first is to write the group as a semidirect product $\mathbb R \ltimes \mathbb R_{>0}$. There is a theory (due chiefly to Mackey) that deals with the representations of semidirect products. In this case, the semidirect is fairly simple, so the description of the unitary irreps you get from "the Mackey machine" is simple too.

The second approach is to realize that the $ax+b$ group is exponential solvable. This means that Kirillov's orbit method applies. Thus the unitary irreps lie in correspondence with coadjoint orbits of the group on its Lie algebra. As a vector space, the Lie algebra of the $ax+b$ group is $\mathbb R^2$, and the coadjoint orbits turn out to be the half-planes $\{y>0\}$ and $\{y<0\}$ and the singletons $\{(x,0)\}$, so the picture isn't very complicated.

A good, modern reference for both approaches is Folland's *A Course in Abstract Harmonic Analysis* (CRC Press, 1995), sections 6.7 and 7.6, respectively. Folland also describes the Plancherel formula in section 7.6. I should mention though that Folland doesn't prove absolutely everything when it comes to the orbit method, but he does give references.