Well having lived my entire life in the shadow of Birmingham I would say avoid living in a suburb of Brum entirely if you can, I know some people mentioned Worcester, which is much nicer ...but you don't need to go that far. I live just outside of Birmingham, west of the last suburb to the south west, which is Longbridge (I think Rednal falls under Worcestershire), just over the border in North Worcestershire, near Bromsgrove and it's much better, I'm not saying it's great, it's not, but it's much quieter and much more peaceful with far less trouble, no homeless druggies, riots or anything like that really. You wouldn't have a problem renting in and around Bromsgrove for your budget, or indeed a little further like Droitwich, the area I am talking about basically falls between junction 4 and 5 on the M5.
If you must live in a Birmingham suburb then I would suggest Edgebaston as being the best option that is really close to the city centre, a little further out is Harbourne which isn't too bad as far as Brum goes and Bartley Green is pretty quiet (my grand parents live there) ...actually Bartley green is on the edge to the West, further than that you cross into the 'Black Country' and Halesowen and to the south into Worcestershire again.
(In answer to your question about Mosley, it isn't too bad, there are far worse places you could live.
Really your best bet is to come over towards my neck of the woods, Bromsgrove. You still fall under the Birmingham post code, it goes as far as Droitwich actually, which is nearer to Worcester than Brum, but it's in the county of Worcestershire and thus under Worcestershire county council (I'm not so sure that's a good thing though as they tend to pinch the pennies far more than Birmingham city council do). Also you don't have to say you live in Birmingham then ...which can only be a plus
