Yes, both sentences are correct. You need a preposition in the second sentence because "go" is intransitive. In comparison "take" is transitive ("take what?"), so "the elevator" is the direct object and no preposition is needed for that.

I see.Thank you, Rachel.

Hi Iris, I would need to correct your second sentence (so it could be British English specific) - I would need 'up in' as follows: 'They went up in the elevator (or more commonly 'lift' in BE). However, you can say 'they went up the escalator' (because you are not inside anything). Rachel.