What's the best reference for actual formulas for RT invariants? If one really wants to understand the formulas for how to construct the Reshetikhin-Turaev 3-manifold invariants coming from quantum groups in terms of R-matrices and such, what's the best reference to read?
Edit:  Apparently I'm being too vague.  Let me explain my motivation a little bit.  Right now, I'm thinking about how to categorify Chern-Simons theory.  I understand most of the maps from quantum groups like the R-matrix quite well, so I would like a good reference that has formulae I can try to categorify based the bits I already understand. 
 A: My guess is that you are mainly interested in going from a quantum group to a modular tensor category.  If so, then this paper by Steve Sawin is fairly explicit and general.
Probably other people (Noah?) have a much better knowledge of the literature than I do.
A: Since no one else has jumped in, I'll try again.
Standard references producing a TQFT from a MTC include Turaev's big book and the shorter notes of Bakalov and Kirillov (available online here).
If you want something less algebraic and more combinatorial (which I doubt), I like the book by Kauffman and Lins, and also some paper(s) of Lickorish's from the early 1990's.
I haven't read the original Reshetikhin and Turaev papers in a long time, but I remember thinking at the time that they were perfectly readable.
A: The formula is hard to implement but not difficult conceptually. The R-matrix formalism gives a link invariant for each irreducible representation. This is then extended to linear combinations of representations. For a 3-manifold invariant you take the regular representation. This means sum over irreducible representations the irreducible with scalar factor the quantum dimension. Then up to normalisation, this is a 3-manifold invariant by Kirby calculus.
