Translation:Take the stairs to the second floor.
Similar to how идите по пешком is literally "go by means of feet", идите по лестнице is "go by means of the staircase." However, the literal translation sounds unnatural, so you should treat it as more idiomatic. "Go to the stairs" would be идите на лестницу, which uses a different grammatical construction.
I think that лестнице is dative ("leading to [somewhere]"). It could be prepositional, but I don't believe it fits the definitions for prepositional objects of по.
In either case, лестнице is singular. Although it is translated as "stairs", it can also be translated as "staircase", which means "the set of stairs leading from one level to another". In English, "staircase" is also singular, though "stairs" is plural.