What are good texts to acquaint oneself with standard asymptotic techniques, particularly as they relate to probabilistic combinatorics?
Philippe Flajolet and Robert Sedgewick's Analytic Combinatorics is the most comprehensive reference, available at http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html. Also useful is Odlyzko's "Asymptotic methods in enumeration" (from The Handbook of Combinatorics) at http://www.dtc.umn.edu/~odlyzko/doc/asymptotic.enum.pdf.
At a lower level than Flajolet and Sedgewick, Chapter 5 of generatingfunctionology by Wilf is a good introduction to complex analytic methods. (Yes, my two answers look very similar. As usual in a big list question, I am posting separate answers so people can vote on them separately.)
At a lower level than Flajolet and Sedgewick, Chapter 9 of Concrete Mathematics (Graham, Knuth and Patashnik) is a good introduction to elementary methods.
If you want to know about quantities which (1) have nice generating functions and (2) depend on more than one parameter, the most thorough guide will be found in the papers of Robin Pemantle. to the best of my knowledge, he hasn't written a comprehensive guide to his work; his SIAM report might be the best starting point.
Flajolet and Sedgewick, Analytic Combinatorics (available for free, if you like, on the (sadly, late) Flajolet's web page.