You can also use træningscentret in Danish.
"Fitnesscenteret" should only be one word.
(I'm a native Dane) Yes! Duolingo is very wrong here
Would "for at" and "til at" interchangeable in this case?
I don't think so. I'm pretty sure "for at" serves as "in order to" in this sentence and "til at" doesn't do the same thing.