We bike all around the neighborhood with our kids for school transportation + commuting to work. When we bike with our kids we do ride on the sidewalks. A common rule is to alert pedestrians with a "Passing on your left!" warning.
Driving is a big headache. Especially because parking is really hard around here (especially if you work at the U of C.) Anyway distances aren't that big to warrant driving.
For preschool age children, I highly recommend a trailer. I had a child's seat on a bike for a while. I found it a bit shaky. Maybe it was installed wrong but it fell off twice (without a child). With a child's seat you want a "women's" because their weight can make the bike wabble a lot. With a trailer, you can protect the child from the cold and ride throughout the year with proper winter clothing and cary school bags + groceries. Burley is the top of the line & the cost reflects that. Trek seems sturdy.
We just got a tagalong for out almost-4-year-old and he loves it. He knows how to ride w/ training wheels.
I did find the university's transportation web page. I have to warn you that those public buses that go around the neighborhood aren't the most reliable.
http://facilities.uchicago.edu/transpparking/index.html
You can ride them free with a Univ of Chicago ID.

Chicagoland Bicycle Federation has more about bicycling with kids:
http://www.biketraffic.org/content2.php?id=27_0_6_0
South Side Bicycle Movement
contact Keith Holt
by e-mail at Keith@biketraffic.org,
or call (312) 427-3325 ext.
If you think Hyde Park could be more bicylce friendly, Keith recommends commenting to the aldermen.

I also found this Chicago city bike ordinance:
Riding Bicycles on Sidewalks and Certain Roadways
9-52-020. Riding bicycles on sidewalks and certain roadways.
(a) No person shall ride a bicycle upon a sidewalk within a business district.
(b) No person 12 or more years of age shall ride a bicycle upon any sidewalk in any district, unless such sidewalk has been officially designated and marked as a bicycle route.
(c) Bicycles shall not be operated on Lake Shore Drive or on any roadway where the operation of bicycles has been prohibited and signs have been erected indicating such prohibition.
(d) Whenever a usable path for bicycles has been provided adjacent to a roadway, bicycle riders shall use such path and shall not use the roadway.
(Added Coun. J. 7-12-90, p. 18634)

I guess that means if you're pulling children you can ride on the sidewalk.