Hello everyone.
The rent is cheaper in Knoxville than most cities in the US, which is really nice. If you want to be really close to campus there are a lot of houses and apartments in the Fort Sanders area. From anywhere in the Fort, it is only at 5-10 minute walk to campus. There are a lot of nice houses, a lot of very livable houses, and a lot of houses that are in terrible condition. Nearly every undergraduate lives in the Fort, so it can be a loud party scene.
If you move out of the Fort, rent gets even cheaper. Assuming you want to be away from the undergrad party scene, a lot of people live in North/South Knoxville. There are not a lot of apartments, but you can rent multibedroom houses on the cheap. These houses tend to have yards and parking, so it is really nice. If you move west of campus, things can be pricier, but there are also a lot of nice options. In my opinion, renting a house is the best option. It can be cheap, and you get so much more than apartment for the same price.
Downtown is starting to get a lot of new loft-style apartments. If you are interested in that, look at Sterchis Lofts and a few others. Downtown Knox is suprisingly fun, so living there could be nice. The 4th and Gill neighborhood just outside of the downtown area has a lot of Victorian style homes that are usually in great condition and rent for cheap. If you search craigslist for housing downtown you will get results for immediate downtown area, the Fort, North Knox, South Knox, and some West Knox.
If I've repeated any information, I apologize. Just wanted to give my input. Best of luck on the search! And welcome to Knoxville. It's a great city with a fun music culture, lots of outdoor activities, and awesome people.