It really depends on what your top end budget is. I lived in Middlewood, just the other side of Hillsborough for a while which was great for Greno and also very accessible in to town on the tram. If you could afford it I’d look around Ecclesall Rosd South around Greystones as there are a lot of rental houses on the likes of Ranby Road.
I’ve also lived out in Hathersage, fantastic village and lovely being out in the Peak but not great in terms of easy access back in to Sheffield for work. Now in Totley which I think has a lot going for it but not many rentals round here, not cheap and still a bit out of town.
Have a look at Grenoside, Oughtibridge, Meersbrook or parts of Sharrow areas as these are cheaper than the west parts of town but still have a good community feel to them. Oh and welcome to Sheffield! It’s reet good.