Going up to something uses the verb monter in an intransitive manner and therefore requires être as auxiliary verb. When you say "going up the stairs" (note: not "to the stairs") "the stairs" is a direct object and therefore monter is used transitively and needs avoir. When used with être, you will always have a preposition along with it. Not so with avoir:

Je suis monté dans la voiture. -- I got into the car.
Je suis monté sur le toit. -- I got up on the roof.
J'ai monté les escaliers. -- I climbed the stairs. (not to the stairs, in the stairs or on the stairs).
J'ai monté ta valise. -- I took your suitcase up.