Dore is expensive but very organised. you name it , it's got club for it - or 2/3. Excellent schools, park, nurseries, activities for children etc. Village feel, edge of Peak district. No chance 4 bed on your budget.
Totley a bit cheaper. Both quite a way from Longley.
South West S generally, pretty but hilly, high up and COLD in winter. If you're to be pushing a pram you'll want to be on the flat and nearer to the centre of these villages, areas etc. Do you like snow? hiking, riding, climbing? all big in S.
Sheffield is relatively compact in comparison to other cities so you will be able to take advantage of activities in the centre when children older.
Try Ecclesall bit nearer to centre than D ( or Banner Cross / Hunter's Bar) good access to shopping at BX HB, Ecclesall Woods, Endcliffe Park, Botanical Gardens etc.etc. Crookes good shops, lots on, Crosspool higher, smaller shopping centre. Fulwood ( pretty, village feel, lovely park, small shopping centre ) - all above excellent schools / edge of PD etc too. Lots on. Nearer to L than D or T.
All ( except HB) in Sheffield Hallam Constituency with one of the highest number of graduates in England, richest C in Sheffield. Left Labour MP Olivia Blake.