do you have a car? Close to campus there is the university circle area. there're apartments at the corner of euclid and mayfield, and hessler road which is practically on campus is an awesome street. Little Italy is a great area (if you map it out on mapquest, little italy is on mayfield road, between euclid and coventry), if you live near mama santa's though, you will be morbidly obese by the end of your first year, their pizza is delicious and ridonkulously cheap. Coventry is another really really cool area. There are a lot of bars/coffee shops on coventry, and the apartments are plentiful and nice. If you live out there though it's going to be a 20-25 minute walk to class.
Shaker Heights is farther away from the school, but still a decent area. If you lived there you'd want a car most likely. Livind downtown would also probably require a car (unless you're really big on public transporation). That being said I do know people who live pretty far out there and just take the bus in or get rides with roommates, so it's workable.