I'm assuming you're interested in complex representations of these groups. In each case the unitary, tempered and generic duals are contained in the smooth dual, so I'll start by describing that. Let $G$ be any of the groups you're interested in. Actually, most of what I'll say goes through pretty nicely for $G$ the group of points of a connected reductive group over a local field, with some modifications to take the more complicated structure into account.
The first key step in classifying representations into the broad classes you describe is parabolic induction. In $G$ there are certain parabolic subgroups $P$, which have a Levi factorization into a semidirect product $P=MN$, where $M$ is another nice group of the type you're interested in (a Levi subgroup) and $N$ is a normal subgroup. When $G=GL_2$, every parabolic is conjugate to either $GL_2$ itself (which obviously isn't going to help), or to the group $B$ of upper-triangular matrices (the Borel subgroup). The Borel subgroup has a Levi decomposition $B=TU$, where $T$ is the diagonal torus and $U$ is the group of upper-triangular matrices with 1 along the diagonal. In $SL_2$ and $PGL_2$, the same picture is true, but you restrict and project these subgroups, respectively.
Now take $\chi$ an irreducible (so one-dimensional) representation of $T$. You can inflate $\chi$ to $B$ by $B\rightarrow B/U\stackrel{\sim}{\rightarrow}T\stackrel{\chi}{\rightarrow}\Bbb{C}^\times$, and then form the induced representation $Ind_B^G\ \chi$. This is a finite-length representation. Subquotients of this induced representation break into families:
- Finite-dimensional representations. If you're working over a $p$-adic field then these are one-dimensional and factor through $\det$; if not then for each $n\geq 1$ there is a unique family of $n$-dimensional representations, where any two in each family are twists of one another by a character through the determinant.
- Principal series representations, which occur when $Ind_B^G \chi$ is irreducible.
- If $Ind_B^G\chi$ is reducible, it may or may not be semisimple. If it's semisimple then it splits as a direct sum of two irreducibles, which are called the limit of discrete series. This doesn't happen over a $p$-adic fieldd. Otherwise, it has a unique irreducible subrepresentation, which is one of the finite-dimensional representations. The quotient by this representation is formed of discrete series representations.
Over the reals or complexes, going through this process gives you all irreducibles. Over a $p$-adic field, there is a family which you don't see in this way: the cuspidal representations. These are much, much harder to construct.
The general picture is similar, but combinatorially it becomes much more complicated as the rank of the group grows. In general there are multiple conjugacy classes of parabolic subgroups, and one has to consider parabolic induction from each of these. Principal series always refers to representations parabolically induced from a Borel subgroup, and discrete series has a more general meaning as well.
Once you know the dual of $GL_2$, you basically know that of the other groups. Given $\pi$ an irreducible representation of $GL_2$, $\pi$ identifies with an irreducible representation of $PGL_2$ if and only if it has trivial central character, and all representations arise in this way. On the other hand, $SL_2$ is a normal subgroup of $GL_2$, and by Clifford theory the restriction of $\pi$ to $SL_2$ splits as a direct sum of pairwise $GL_2$-conjugate irreducible representations (moreover, each appears with multiplicity one). This restriction to $SL_2$ preserves containment in each of the above families of representations.
