A few months ago I posted similar on here and kept a note of all the responses. It's a bit long but here's what the lovely MNers told me:
Good areas of Birmingham:
Bournville
Castle Bromwich (village)
Edgbaston
Erdington (i.e. around Orphanage Road/Grange Road and Pype Hayes Park)
Great Barr (Red House Park/St Margarets)
Hall Green (parts of)
King's Heath
King's Norton (parts of)
Harborne
Moseley
Selly Park
Stirchley
Sutton Coldfield (especially in Four Oaks and Mere Green)
Sutton Four Oaks
Sutton New Hall
Sutton Trinity
Sutton Vesey
Bad areas of Birmingham:
Aston
Bordesley Green
Handsworth
Handsworth Wood
Highgate
Ladywood
Lozells
Newtown
Saltley
Small Heath
Sparkbrook
Washwood Heath
Winson Green
Nearly everyone has mentioned, Handsworth, Aston and Lozells etc., in fact some taxi drivers won't drop you off on Lozells Road, as they say it's too dangerous, and they have a point.
But there are lots of nice places as well. Erdington is quite rough, but there are some nice areas (near Pype Hayes Park), also Edgbaston, Moseley and Harborne are nice.
Sutton Coldfield is a nice area, especially in Four Oaks and Mere Green. However, there are places in Sutton you need to stay away from, like Falcon Lodge
Edgbaston and the private estates in Sutton Coldfield are the wealthiest areas of the city; here you will find sprawling houses with tennis courts and swimming pools. Next in the hierachy are the leafy and interesting suburbs of Harborne, Moseley, Selly Park and Bournville - all concentrated in the professionals' haunt of south Birmingham. Aside from Sutton Coldfield, parts of Erdington (i.e. around Orphanage Road/Grange Road and Pype Hayes Park); Great Barr (Red House Park/St Margarets); and Castle Bromwich (village) are very pleasant areas in north Birmingham.
Further afield, other nice areas include King's Heath, parts of King's Norton, parts of Hall Green and the leafier fringes of Solihull (technically not in Birmingham but another very affluent area).
The rest of suburban Birmingham is rather a hotchpotch of good and bad; working class residential areas intertwined with council estates.
Don't go anywhere near the inner city areas. Unfortunately, Birmingham's inner city is now frighteningly ghettorised. These areas include Newtown and Lozells, Handsworth (and sadly now Handsworth Wood), Winson Green, Highgate, Ladywood, Sparkbrook, Small Heath, Saltley and Aston.
There are 40 wards in Birmingham (A map can be found on the Council's website). I'd break these 40 down as follows:-
Dodgy = Aston, Bordesley Green, Washwood Heath, Kings Norton, Kingstanding, Ladywood, Longbridge, Lozells and East Handsworth, Nechells, Perry Barr, Soho, Sparkbrook, Stockland Green.
Not as Dodgy = Northfield, Shard End, Springfield, Stechford and Yardley North, Weoley.
Some ok bits = Acocks Green, Brandwood, Erdington, Handsworth Wood, Hodge Hill, Quinton, Sheldon.
More ok bits = Billesley, Bournville, Edgbaston, Moseley and Kings Heath, Oscott, Selly Oak, South Yardley, Tyburn.
Mostly ok (some Dodgy bits) = Hall Green, Harborne.
Ok = Sutton Four Oaks, Sutton New Hall, Sutton Trinity, Sutton Vesey.
South Birmingham generally is nice, a surprising amount of parks, good schools. Kings Heath is probably the most Bristoly of the suburbs (from what I hear of Bristol), a bit alternative, cafes, although the high street is very normal with Poundland etc. Stirchley, where we live, is an odd place, run down, desperate for some money and development, but has a good community, including newcomers who are very committed to it. It's also close to the canal and the River Rea cycle route which are great for cycling to the Uni (canal goes directly there) and to town, and out of the city, if you're into cycling. There's a history of co-ops in the area and two have opened over the last year. I expect as it becomes more middle class it'll change but I think quite slowly as there's no money atm.