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I have heard excellent things about Hovey and Hirschhorn (Goerss-Jardine notes in Ch.10 of the second edition that the latter is the standard reference on localization). Is it worth reading only one of them, or should I read both of them, and if so, in what order?
I would read Hovey first, then Hirschhorn.
Does either one go over the double category approach that May says you told him about in ConciseII?
No, the only references for the double category approach are ConciseII and the original paper (hmm, the arxiv version of that is out of date, though).
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