If we define a quantum group to be a quasi-triangular or coquasi-triangular Hopf algebra, then what are the major families of quantum groups?
Of couse, to start with we have the h-adic completions of the Drinfeld-Jimbo deformations of the enveloping algebras of complex semi-simple Lie algebras $U_q({\mathfrak g})$, and the dually defined quantised coordinate algebra of the compact semi-simple Lie groups. With these two families really being two sides of the same thing.
Apart from this, the only other main family that I know of are Majid's bicrossproduct Hopf algebras, which he says can be thought of as deformations of solvable groups.
Majid has two other constructions called bosination and double-bosination, but I don't know if these are quantum groups in the sense defined above.
Best Answer
I've been having trouble answering this question because I think your notion of "quantum group" is either too restrictive or too expansive. Hopf algebras suffer from annoying analytic issues as soon as they're infinite dimensional, so you should either be looking at finite dimensional quasitriangular Hopf algebras, or you should pick some particular world of analysis you want to work in (C* quantum groups, h-adic quantum groups, etc.). On the other hand, there's no real reason to restrict your attention to Hopf algebras, lots of things that go under the name "quantum group" (most notably the semisimplified categories at a root of unity which occur in Reshetikhin-Turaev's construction of 3-manifold invariants) are not the category of representations of a Hopf algebra, instead they're a braided tensor category.
Anyway some important constructions that you don't mention include: