The law on that's a bit nebulous.
The revelant statute is 720 ILCS 5/ Criminal Code of 1961, Title III, Section C, Paragraph 19. It reads:
(720 ILCS 5/19‑2) (from Ch. 38, par. 19‑2)
Sec. 19‑2. Possession of burglary tools.
(a) A person commits the offense of possession of burglary tools when he possesses any key, tool, instrument, device, or any explosive, suitable for use in breaking into a building, housetrailer, watercraft, aircraft, motor vehicle as defined in The Illinois Vehicle Code, railroad car, or any depository designed for the safekeeping of property, or any part thereof, with intent to enter any such place and with intent to commit therein a felony or theft.
Possession of burglary tools in violation of this Section is a Class 4 felony.
(Source: P. A. 78‑255.)
As you can see, it covers any
instrument that can be used in breaking and entering. It also specifies "with intent to enter". These mean that if you're caught with picks (or a screw driver, brick, sledge hammer, etc.) while either actually breaking into somewhere (or just loitering around and acting in a way that a cop could deem suspicious) then you're in felony teritory. If you're just carrying your picks and the police have no reason to suspect you of anything you should (hopefully) be fine.
Personally, I have a few sets of picks that I carry on my person and have in my tool kit at home, and have never once had a problem. If you limit yourself to the hobby aspects of lock picking, you should be just fine. If you do buy picks, I'd also recommend picking up a lock or two at Ace or Menards to practice on. That way there will be less of a temptation to try and pick licks that are in use, and you can carry your practice material with you.
Good luck & easy picking,
No matter where you go, there you are. - Buckaroo Banzai