Jess,
Please bear in mind that I'm some 20 miles away and have never lived in Norwich, so my impressions which follow are purely from being a regular-ish visitor to the city. Just so you don't put too much stock on anything I might say!
I can't actually picture Quebec Road. I'm usually just on the major thoroughfares when driving through there -- Ketts Hill, Plumstead Rd. etc. I have been through some of the side streets in that neighborhood a time or two though, possibly even Quebec Road itself. The lower rent compared to further out sounds about right. As far as I can recall there are quite a few streets around there with rows of old Victorian houses, with some 1930s and later bungalows thrown in between for good measure. Overall, the feeling is of an older and more densely built-upon district than you'll find in the houses around the school at Thorpe St. Andrew a couple of miles away. In fact the old gas works are between Quebec Road and the river, which gives you an idea of how well-established a part of the city it is. In other words, don't expect anything like American surburban spaciousness!
I asked somebody I know who is Norwich born and bred about the area, and she's never heard of any particular problems around there. As she said, being where it is it's not one of the quietest neighborhoods going, and not as nice as Thorpe St. Andrew, but nothing really bad to report.
If you look at the Google map you can get an idea of what's close by. Follow Riverside Rd. down to the bridge and the intersection with Thorpe Rd. Norwich train station is right on the southeast corner of that intersection. Go across the bridge to Prince of Wales Rd. and you're in one the main shopping streets. It's only a few minutes walk from there (either up Prince of Wales Rd. or Rose Lane) to Castle Mall, a multi-level shopping mall. The central market place isn't too far away. I'd say that it's all within walking distance, so long as you don't come back too loaded!
Back from where we started, if you go up to Ketts Hill and then turn west to the roundabout at the north end of Bishop Bridge Rd. you're at the bottom of Mousehold Heath, a quite extensive green area which sits on the hillside. Gurney Road twists its way up through the heath where you'll find fields for weekend ballgames, woodland trails, and so on. Again, much is probably within walking distance if you're fit enough to get up the hill (Brittania Road which goes past the old jail will take you there as well, but is quite steep in spots!).
I hope that gives you at least some idea of what's around there. Trying to imagine a place just from a map can be quite difficult.
I'll probably be going into Norwich later in the week. I'll see if I can swing by the school and Quebec Road area and get you a few pictures.