I definitely recommend that you visit, because a lot of people have a view of Sheffield as a very urban place. However, much of the city is much more like a load of small villages sort of connected together with big belts of greenness inbetween. It's far more like a big town to live in than an urban experience. There are some places where you would honestly think, looking out of the window, you were in the middle of nowhere, yet you are actually 20 minutes by bus from the centre.
A (very subjective) rundown of areas.
Basically, to the south/west, the city is bisected by three arterial shopping roads: Ecclesall Road, Abbeydale Road, Chesterfield Road. Ecclesall Road is slightley posher, and increasingly dominated by big chains, though there are still loads of lovely local shops and restaurants. Abbeydale Road is a world of wonderful bonkersness, largely independent shops, with an increasingly hippy/hipster vibe. Chesterfield Road is up and coming - lots of shops here have been closed for a while, but it's increasingly become a destination for pubs and food with some nice new bars and eateries opening up. The areas around these, particularly towards the edges of the city, are very green - if you looked at them on a satellite view, you'd see tons of trees.
Directly to the West of the city centre you have:
-Broomhill - posh area, close to the university. A real mix of very wealthy permanent residents and buy-to-let housing for students. Can be noisy after dark, but is very close to the city.
- Crookes: studentsville! Lots of terraces, some good local shops. Housing tends to feel more pennine, because it's stone not brick.
-Walkley: slightly posher, high up, slightly further out than Crookes, convenient for the uni. Many academics and postgrads here. A mix of terraces/semis, tends to be fairly denseish.
-Crosspool - probably the area of Sheffield I know the least. Further out than Broomhill, seems to be lots of 30s semis with the odd pub/local shop.
Then, working west of Ecclesall Road, you have Fulwood - this is a weathy suburb with a few independent shops. Big houses, a mix of modern and old. Feels really close to the countryside. Slightly further out is Lodge Moor where you really are right on the edge of things - massive views from some properties here across some beautiful valleys, but good transport links to the centre. Both areas are quiet, green, settled family areas but - forgive me anyone who lives there - a bit boring if you want to be part of the scene in the city 
.
-Ecclesall Road/Endcliffe Park area: wealthy, posh, but closer in - and highly desirable, so one of the most expensive areas. Larger houses, close to the Botanical Gardens. Rustlings Road, near the park seems to be highly desirable.
- Nether Edge: lies between Abbeydale and Ecclesall roads. This is full of big Victorian houses, and tree-lined streets. There's a wealthy, slightly leftie vibe to it, and a strong middle class community spirit. Gardens tend to be smaller, and housing is relatively dense- grand terraces and semis.
-Sharrow/Hunters Bar is the area next to Nether Edge and closer into the centre. It's got slightly smaller houses (think rows of traditional norther terraces) and a less green, more urban feel. Sharrowvale is now brilliant for posh shops and there are a lot of great places to go after dark. There are nice pubs, restaurants, delis, art shops, etc. Most postgrad students I know live around here.
- Greystones: further out than Nether Edge, more late C19/early C20 terraces and semis. Higher up, good views, liked by younger professionals and academics.
- Carterknowle is further out than Nether Edge, and more suburban in feel. It's affordable, and in a good catchment area for schools and on regular bus routes to the university. You could probably get a semi here in your budget. Greystones is across Ecclesall Road and similar in feel - it is perhaps a little more terracey, but feels very well-to-do.
- Woodseats is further to the south than Carterknowle, and is a combination of grander semis, 30s semis, and terraces. It's highly affordable, a mixture of older families who have lived in Sheffield all their lives and younger families who are increasingly moving in because it's cheap yet close enough to brilliant pubs and restaurants to walk out for an evening.
- Meersbrook: an increasingly interesting area of shops is opening up along Chesterfield Road, with bars, pubs and restaurants. It all feels quite excitingly provisional at the moment - lots of artists/writers/young professionals have moved into this kind of area and it seems to be a reflection of a slightly hippified, alternative crew. Meersbrook Park has great views and some very nice housing around it. Housing is quite dense, however, so getting huge amounts of green space is difficult.
-Heeley: the more urban end of Meersbrook, quite socially mixed but very friendly in feel. There's a brilliant city farm, but green space attached to houses tends to be more limited.