Reference for Quantum groups

Brain dump: Kassel is good. Jantzen (Lectures on Quantum Groups) is nicely written and clear, it's my favourite. Majid (Foundations of Quantum Group Theory) is an option, I haven't read it. Lusztig's book has a reputation of being tough to read. Chari and Pressley (A Guide To...) is comprehensive and has some interesting material on knots, 3-manifolds, the KZ equation and the absolute Galois group.

Our (Carter, Flath, Saito) "Classical and Quantum $6j$-Symbols," PUP does not cover the subject in general, but tells a lot of what is needed in the $U_q(sl_2)$ case.

For starters you can check references in this question of mine on applications of Hopf algebras.

Here's an article or two that build a little theory of ${\mathfrak sl}_2$ related stuff. There are many similar articles on spin-chains where ${\mathfrak sl}_2$ deformations are naturally present.

References in the classic Di Francesco, Mathieu, Sénéchal on conformal field theory.

Another connection (via Yang-Baxter equation) is with integrable systems.