Fly fishing?

When fishing a dry fly on a stream it is of great importance that the fly is not allowed to drag, but what about when fishing wet flies and nymphs? I have come across conflicting sources of information on this; some say that drag should be avoided, whereas others say that they should be allowed to drag, or that it does not no harm at least. Could someone please clear this confusion up for me? Also is it best to fish them casting upstream or downstream?