My son is a gifted student who has always tested at the top of the test ranges in all subjects and was ahead of many class subjects in 1-3rd grades. He doesn't go to a CPS school but they don't start pulling for gifted until 4th grade at our school.
Certainly I would say monitor your son's progress at school through their regular testing proceedures, etc. but don't rely on a 'gifted' program this early, as many schools just don't do them at that age. Encourage your son to read at his own level and work at home with him on flashcards, history or science projects, art and music, etc.
There are some very good reasons many schools don't do 'gifted' classes in 2nd grade or even 3rd. In third grade, kids start to get more 'real' homework' and that is also the year many of them start a regular course of social studies and science, more formalized than things they do in earlier years. There is a LOT to absorb socially and structurally in 3rd grade, even if your son seems academically ahead of other kids now.
Now in 4th grade, my son is pulled out for gifted sessions a few times each week, learning latin roots, and other things. However- he still has the same organizational skills as most 9 year olds! Even though he can DO all his homework easily, there is a lot more of it and just learning to be organized, keep track of things, and schedule himself is a big part of what 4th grade is for. In 3rd and 2nd, I remember worrying if he ought to have gifted classes then, but now I am VERY happy that they waited until he had a little more experience under his belt.
Enrich your child at home and monitor his progress with the school, but don't push him too fast, too young before he is mature and organized enough to succeed- you want gifted classes to be a challenge and interesting to him- not something that overshadows his early school experience.