(affordable neighborhoods that aren't populated by undergrads, near public transportation)
There are affordable apartments in almost every neighborhood (save Old Town or the South Loop I suppose), but you should definitely investigate Logan Square, Avondale, Ukrainian Village, Edgewater, and Lincoln Square. Pilsen is popular students. I had a decent studio in Lakeview when I first moved here for $700/month and rents have gone down since then.
Good luck! Let me know if you have questions about specific areas...