Picking up Chicago missions is a pretty decent way to go. Just know that the final four (from season 8 ) have not been published -- at this point I have my doubts that they will, what with the focus on 6th Edition.
What you might find helpful -- to see the missing adventures and to see the broader story arc in one place -- is to pick up the Anarchy book Chicago Chaos. Anarchy was designed as a rules-lite version of SR, and they re-packaged Chicago history, (most of?) the Chicago missions (and maybe a few extras?) and some other material for Anarchy.
A few caveats:
- Chicago Chaos gives a very summarized version of what is in missions (a page or less per mission)
- Obviously you can't use the character and critter stats from the Chicago Chaos book, as different rule systems. But it should give you a general idea
- some details are changed in Anarchy (they cut out most of the fixers and have characters contacted directly by the Johnsons, for example)
But personally, after having run some missions in my home game, I have found Chicago Chaos to be very helpful.