I moved to Sheffield to study in 2002 and have stayed (well I’m now just south of Sheffield).
As others have said, Sheffield is really a collection of villages (some pretty grim, others really very nice indeed). It lacks the infrastructure and commerce of places like Leeds and Manchester which can be seen as a good or a bad thing. I love being so close to the peaks but do feel quite constrained by the lack of work opportunities in my particular area of IT dullness. Having said that, Manchester is just about commutable by car from the west of the city (the nice bit) and from the centre by train.
Re riding Sheffield offers the below all within 30 mins of most of the city and as others have said you can ride out of some areas straight into the Peak(s)
Dark Peak
White Peak
Wharncliff downhill
Grenoside downhill light
Parkwood springs
Lady cannings plantation
Endless roady-ing into the peaks
No experience of Chichester but it don’t seem lumpy enough for biking down that way imo :-)