"Gehen Sie die Treppe hinauf bis zur ersten Etage", to keep it as close to your text as possible, is correct German. Note that this is what you would say in a detailed decription that covers more than just getting to the second floor. An example would be "Gehen Sie den Gang entlang bis zur Treppe, dann gehen Sie die Treppe herauf bis zur ersten Etage, dann gehen Sie durch die blaue Tür..." "Gehen Sie die Treppe hinauf in den ersten Stock" is also correct. Note that here, the second floor is a destination. In the previous sentence, it was the location where you stop climbing the stairs. This is the distinction Cherry is pointing out. If you only want to tell someone that he should go to the second floor, you would just say "Gehen Sie in den zweiten Stock" as suggested above. (And you would not mention the stairs unless there are other equivalent means of reaching the second floor.) Please note that what Cherry wrote about dative and accusative in German is only true for prepositions that allow both, such as "in". In these cases, dative indicates a current location (where what the sentence describes is happening) and accusative indicates a destination (that is approached by something as described in the sentence). The preposition "zu" (and thus the contractions "zur" and "zum") require the dative. Unfortunately there is no simple rule that always works, but in many cases, when you use "zu", what you are actually saying is "go to the location of ..." or "go to a location near...", or as in your sentence, "go up the stairs until you reach...".

Hello Laura, Notice you say as mentioned above: " in den zweiten Stock " Here you used the 'Akkusativ' to express movement , from where the person stands downstairs going up to the second floor. If you used the 'Dativ' instead you would say " im zweiten Stock " and it would give another meaning then " on the second floor" . I know the 'Dativ/Akkusativ' issue has been confusing you many times and this hint is just to emphasize on how using both 'Akkusativ' and 'Dativ' do change the meaning drastically one indicating a moving state "Akkusativ" and the other a static one "Dativ".

Es heißt: in den ersten/zweiten Stock gehen.

Please note that the german first floor (erster Stock /erste Etage /erstes Obergeschoss/OG) is the english second floor. Your first floor (ground floor) would possibly be "das Erdgeschoss" (EG).

in den zweiten Stock auf die zweite Etage ins zweite Obergeschoss