For the University of Warwick, Coventry is indeed the closest town, but there may be better areas. That's not to put down Coventry as there are some very nice areas close to the University, but they're mainly large private homes.
Writeshawnna has mentioned Kenilworth. It's a 'smaller town/larger village' and is quite pleasant. Fairly good shopping, including Waitrose (food shopping), Boots (chemist/drug store), W H Smith (cards, magazines, and more), and many small branches of most UK banks. It also has it's own castle ruins (see Kenilworth, Sir Walter Scott). Not the most scenic Warwickshire village, but was one of the first places I looked for accommodation when I moved to the UK (my first employment was in Coventry).
Balsall Common is also close, but it's smaller and is really a bedroom community for Coventry and Birmingham. It is a pleasant community if something is available.
Leamington Spa is the largest town outside Coventry, and has student accommodation due to Warwickshire College being located there, but it's still probably only 5 miles from U. of Warwick. Buses to U of Warwick, I assume, are available. (Many buses to Coventry.) Warwick itself and Leamington Spa are 'joined at the hip'. I'm not sure which you would call the suburb of the other, but both are worth a look if transportation isn't a problem. Warwick (with its castle) is a bit more scenic.
