I’ve actually put a lot of painstaking work into a personal reading order for myself over the past several years, mainly focused around the JSA and several other characters/series that I’m interested in reading. I’m happy to share an abridged version with you. This runs from cover date April 1999 (JLA #28) to October 2011 (Justice Society of America #54). If you want anything before or after that, just let me know.
I’ve composed this list in such a way that individual storylines are not broken up while jumping back and forth between series, but everything should be concurrent. In other words, unless it’s a crossover storyline directly running between two series, you’ll mostly be reading a trade at a time while reading multiple series at a time if you use my list (if that makes sense).
I have included event series that the JSA are involved in and which cross over with the various JSA series; feel free to read any, all or none of these. Some (actually a lot) of the things on the list aren’t available on DCUI, but I’ve included them for completeness’ sake; I have denoted these series/issues. If you’re really dedicated to reading everything, you may be able to track them down elsewhere. As far as I know, everything here should be in the right reading order, but if I’ve made any mistakes or left out anything that should be there, please let me know.
P.S. I have a much more extensive version of this list that integrates related series that ran at the same time, like Hourman, Stars and S.T.R.I.P.E., Power Girl, Magog, etc.; if you’d like that, just let me know.