I haven't seen many good strength gyms outside of Chicago itself, but this one in Lombard looks really legit and it is pretty easy to get to:http://goldenagestrengthclub.com/ It's not a starting strength gym, but the owner is pretty bad ass and appreciates powerlifting so I'd imagine it would be do-able.
I'm not yet a member of this gym because a) it costs $50/month and b) my weak lifts can be done at my $19.99/month gym. However, I am pretty excited to join this summer, it's like a crossfit gym but you can do whatever you want by yourself.