R.
The only person I know who lives in Hyde Park (wonderful area) sends their child to a private school. You are going to have to do a lot of research before moving--so you are off to a good start! The Chicago Public School system can be very tricky to understand--there are differences between just regular neighborhood schools and magnet/charter schools(which have lotteries and/or waiting lists). Some schools are excellent and others are not. We moved out of the city because I found this process to be exhausting and just wanted a school that my daughter to could walk to, but you need to have A LOT of money to afford a home in a nice neighborhood--and we could not. Most of my friends live in the city, but right now their toddler children attend private schools.
There are lots of great schools throughout the city (I mostly know of ones in Lincoln Park and Lakeview--North Side of the city), just be prepared for applications and waiting lists!
Good luck!