Kelham island/Neepsend/West bar for central flats. Kelham island is pretty trendy, Neepsend is its up-and-coming cousin... I mean fuck off Neepsend tbh, it's where my workshop is and I don't want more flats there! But yeah, that area is popular. I live in West Bar, which is near there, marginally closer to centre, mostly flats and student blocks. The area has a ridiculous number of very good pubs.
Crookes is more traditional student - shared houses as opposed to blocks of flats. Walkley is... Slightly outside Crookes... And I suppose represents post-student stuff. Nice terraced houses. Nether edge is the other way and further out, bit quieter, quite expensive, but not as snobbish as Fulwood. See also Ecclesall Road/Sharrow/Endcliffe park (you can actually walk up to the peaks via Endcliffe park).