I've been in Chicago or the Midwest since 2009 now. Definitely go for public transit if you can. Parking in the city is ridiculous (cost and availability) and it's more hassle than it's worth. Public transit/walking should be your first choice--it'll save you so much money and hassle. No scraping the windshield, no shoveling your car out, no shoveling snow (if you have a duplex or apartment that requires you to clear the driveways), etc.
If you do need a car:
A remote starter is nice, but honestly not required if you can just walk to your car and start it--this assumes you're renting a house/duplex with a private garage or driveway where'd it'd be safe to leave it running with the keys in it. Obviously the remote starter would help on your way home from work, but honestly I just get in my car and start driving. If I'm running late I do that in the AM too--it's no big deal. Half of Midwesterners don't warm up their cars, and half of mechanics say that's a bad thing to not warm up your car and half say it's what they do and it doesn't matter (you get your car up to operating temperature fairly quickly).
You do not need a 4x4 (they're usually really just meant for off-road and deep snow--you can actually wreck most 4x4 systems in light snow/rain because they often don't have a differential between the front and rear axles). Still, that doesn't stop my sister from using her Jeep in 4x4 in rain in CA--her Jeep seems fine so far and it doesn't have the differential...
FWD with snow tires is far better than AWD or 4x4 with all-season tires. AWD and 4x4's are expensive. An extra set of rims (look on Craigslist) and snow tires should be around $400-$800. Even in a place like Chicago where it's rare that you'll be driving on top of snow except during the storm itself (they clear the snow up fast), it's still frigid, and snow tires don't turn to rock in cold weather like all-season tires. I have snow tires for winter with a RWD pick-up and it's done wonders, even when I lived in a smaller town where the side streets seemed to never get plowed until 2-3 days later... Remember, all-season tires really mean all seasons you'd encounter in Los Angeles or Phoenix.
Leave for work early if a storm is expected--a bad storm in the middle of commute can easily triple the commute. So pay attention to weather reports, which will often be wrong anyway, but it's better than nothing (and usually when the big stuff is coming they're fairly accurate). And as someone mentioned, consider just staying at the hospital overnight. I never had to in my 6 years so far of living in the midwest, and I think I've only had a total of three bad snow commutes.
It's really not that bad. I lived in CA all my life, and in SoCal they shut down route 5 (the biggest freeway connecting Northern/Southern CA) for half an inch (or any really) of snow. You really learn to toughen up with the winters here (shorts and t-shirts in March when it's back in the 30's). It's a blast--the people are great here, and Chicago really is a great city.