Lurie: Excellent course on stable homotopy theory: http://www.math.harvard.edu/~lurie/252x.html
Classically, the stable homotopy category is the homotopy category of the model category of spectra. Replacing spectra by symmetric spectra or S-modules give equivalent results.
Hovey, Palmieri and Strickland discusses axiomatic notions of stable homotopy categories. For more on this, see their original article, the discussion in Hovey’s book, section 7.2, and also the article in Greenlees (ed): Axiomatic, Enriched and Motivic Homotopy Theory. A related MO link: http://mathoverflow.net/questions/98982/non-noetherian-stable-homotopy
To get a flavor of the axiomatics, Hovey defines an algebraic stable homotopy category as closed symmetric monoidal triangulated category together with a set of small strongly dualizable weak generators.
nLab page on Stable homotopy category