I'm based in Pudsey and I used to live in Bramley, so if you want any specific advice for those areas I'll do my best to oblige. :)
I'd agree that if you have the budget to avoid Bramley, I'd suggest doing so. It's not diabolical by any means, but there are nicer areas. There also seems to have a bit of an arson problem, as last week a former warehouse was set alight and I think it was about seven years ago that the doctors surgery was burned down.
Both done by groups of teens.
I like Pudsey, but its not to everyone's tastes. It's got a large OAP population so tends to be on the quiet side, which I prefer but others might not! Good facilities in terms of supermarkets, cinema, leisure centre, library, post office, park, public transport, etc. Easy commute to Leeds; by car Kirkstall would take 15 minutes on a good day and 30 minutes in rush hour.
Horsforth is nice, but be a little wary about skirting too close to Headingley, as there's a huge student population there for the university. Armley has had a lot of investment in it recently so has some shiny new facilities, but parts of it are still quite deprived. It's also where the prison is based.
Calverley might be another place to look at, if you prefer something a bit more green and remote while still within a close commute?
If your budget is reasonable, then the area between Leeds, Harrogate and York is known as the 'golden triangle'. Very nice, but with a price to match.