I actually like the area around the capitol. Maybe not ideal for price, but I think the neighborhood is nice. If I were you I think you would like the area around old university. Bonus, I grew up in that neighborhood (so maybe I am biased). If you are looking at google maps find where university splits from campus drive to the west side of campus near Camp Randall and Engineering. That is "old university rd" the neighborhood is actually university heights, but I don't think anyone calls it that. Anyway mostly upper classman live there. It is a cool neighborhood because most of the houses south of Kendall Ave are residential, but that whole stretch of university where it breaks off from campus drive until it reconnects with campus drive is almost all students. So it is quiet and given that it is about a 15 min walk to campus most underclassman don't live there. Plus I don't think rent is bad there.
You could also look around main st, doty st., basset st. and bedford st near lake Monona. Don't go too much farther east (towards the capitol up the peninsula) because then it starts getting ridiculously far away from buildings you will have classes in (30 min plus walking). Hope that helps!
Also thanks for offering help about Chicago, but I am not an economics prodigy so there is no way I could get into Chicago even if I tried lol.