So you're looking for a 3 bed terrace, with good links to the main hospitals, and with a good secondary school?
A budget of £600 per month may get you a property in Sharrow, maybe Banner Cross.
I would avoid the main student areas - they can get very noisy (so Ecclesall Road, Broomhill, Broomhall, Crookes, Walkley) and very busy traffic wise.
The main hospitals in Sheffield are The Hallamshire, The Northern General, the Children's, Weston Park (a specialised cancer hospital), plus a number of small private hospitals.
I think the secondary schools in order go Silverdale and Tapton. These are very popular, and it's unlikely you'll get a place as demand is very high.
After that you could go for High Storrs, King Edward VII, King Ecberts and possibly Meadowhead.
Or if you wanted to go a way out you could try Chapeltown or High Green. These have a bit of a village feel to them.
I'll have a look and post some links.
You may be able to pick up a bargain youth, as loads of new student accommodation has been build around the city centre, so there are loads of empty properties atm.