Didsbury is nice but it's pretty expensive and a slightly older crowd. Would highly recommend Chorlton though. Lots of decent places to eat and drink etc Ignore anyone who says it's full of c*nts. There's a general trend in the UK at the moment to call anyone who has a beard and enjoys a decent pint a c*nt but it's not actually true.
Whalley Range is a little cheaper and closer to the city centre than Chorlton. There's less going on than in Chorlton, but there are some nice leafy streets there and a couple of half decent bars.
I would give Salford a miss if I were you, even the less scary bits closer to media city are lacking in character and much for a 19 year old to do.
Of the cheaper options, Hulme has improved a lot in recent years and is only a 10 minute walk from the city centre. Also it's just a stones throw from the universities so there's plenty of young people about. Might be good if you're looking for a cheaper option. I lived there for 4 years and had a great time.