Oklahoma native, our cold weather pales in comparison to Chicago - there's a concert in early February right outside of O'Hare that I am headed towards. Most Chicago tourist guides I've encountered online simply don't advise to visit but my hands are tied.
What local things would you recommend to an outsider braving the outside? Any tips on staying warm?
Pretty standard stuff imo, keep your extremities covered; gloves, hat, waterproof boots, and a nice jacket. You'll be fine imo. If you're driving and there's snow on the ground, just take it nice and slow. Accelerate slower than you normally would. Start breaking earlier than you normally would. And plan for travel to take longer when it's snowing.
Maybe some delays, but we're used to snow. There's pretty solid infrastructure for dealing with snow. We have plenty of plows and salt trucks, people generally know how to drive in it, you should be fine taking public transit as long as you bake in a little extra time.