I’ve never heard of the Chicago style tuning. I think that is just referring to what we are all using when taking a standard diatonic harmonica. Richter tuning.
If you are going to get into tuning, first it’s great to fix a harp that is slightly out of tune. Like i show in the videos in the tool shed.
Once you have done that successfully a couple of times, you may want to try and make a difference of a semi tone to a reed, which is the most I’d recommend. In that case, and if you like country music, a fun one to ‘try’ would be country tuning. To do that the 5-draw reed is tuned 1/2 a step higher (to F# on a C harp) giving you a complete major diatonic scale in second position.
Hope that helps, and let us know how you go!