I don't know that I would call it an expression - more a phrase. Up a flight of stairs (or up two or three flights of stairs). The opposite of course is 'down a flight of stairs'. It is used as a way of giving directions - go up the flight of stairs at the end of the concourse and turn left, or, describing a location. The office is up two flights of stairs from the entrance. You can leave the 'flight of' out, and just say, go up the stairs. To me, a 'flight of stairs' suggests stairs which are at least sufficient to get you from one floor level to another. Whereas only a few stairs - say three or four, would be called 'steps', rather than 'stairs'. I have no idea if that makes it clearer for you or not - I hope so!

Yes, in Britain we have 'flight(s) of stairs, and set(s) of step(s). Steps would take you up to a split level, or a front door etc. If you go to a full floor or more it is a staircase or stairway or a flight of stairs. This is relaxing here, and you do occasionally hear set of stairs, but this is less common.