There are apartments on, or adjacent to Newton Road (Valley, Lincoln) that are between the medical school (MERF) and the basketball arena. This area is popular with medical, dental, pharmacy, nursing and law students and employees of the hospital. You are about a half mile walk to the MERF and about the same to the hospital. If you ride your bike, you can literally go home for lunch. These apartments tend to be more costly because of the proximity to the school and hospital.
Another very popular option is the Aspire at West Campus apartment complex. Nice, newer construction. Similar to the modern college town apartment complexes in my undergrad town - with many amenities. Only downside is you need to take the campus bus as it is too far to walk to the MERF. Still on the pricier side due to new construction and amenities.
If cheap is your overarching requirement, there are numerous less expensive apartments or houses to rent in Iowa City. The downside is you will need to take the longer commute into account. By the way, don't think of driving your car and parking across the street from the MERF - you will pay the full daily rate. Just remember, you will likely be at the MERF or library studying late into the night...