I grew up in Chicago and lived there until my first year of college.
It is possible to not have a car, but, if you're going to be traveling at various hours of the night, or in bad weather, sometimes it is really nice to have one. You're going to want to go to the suburbs sometimes too, because as large as the city is, you still can't find everything you need/want within city limits and CTA service areas.
Same goes for the friendships you make and the parties or events you want to attend. Half of those folks will be suburbanites, and not close suburbs either, but places like Naperville. So, consider a car. An old Honda or Toyota could be a very reliable and cost-effective choice. At the very least, consider Zipcar.
The best website I've found out for moving advice is the forums on city-data.com. You'll get great responses to your questions. I've been a member on it for probably five years. Check it out.
In a nutshell, if I were you, I'd consider the South Loop if you wanted something modern/downtown-ish/trendy and you want to live near others who are new to Chicago. If that isn't a priority, consider Bridgeport, which is a close-knit south side ethnic city neighborhood that is safe and interesting. It'll be mostly locals there. You can get a lot more house, or apartment space, for the money if you pick a real Chicago neighborhood.
Since you're going to be in medicine which I'm guessing will be high stress and long hours, I don't think the north side would be a great choice for you because of the distance. It's possible, and I love those areas, but the commute would be tiring.
It is too bad the neighborhoods west of Hyde Park along 55th street are not in better shape. When my parents and grandparents lived in the city, it was safe to take the Garfield/55th bus all the way down to the lakefront and those neighborhoods were thriving. Now those areas are among the worst in the city. If you go due west from U of C, you won't find a pleasant neighborhood until you get to Midway airport.
Good luck with your move!