Translation:I will go home and take a warm shower.
It may also have been marked 'incorrect' because of the different tenses in the translation. Have you tried using the same tense for both verbs? E.g.: "I am going home and taking a warm shower"?
As for the -て form; this is not only for actions currently taking place (having started before the speaker mentioned it, and continuing thereafter) but also for actions that took place in the past - of which we can currently see the result/effect - and for connecting clauses of a sentence. The last verb is always the one that determines the 'time'. In this case, あびます indicates either a present simple or pseudo-future tense.
Ok this is absolutely nonsense, this might be my english failing here but, あたたかい means 'hot', which is why I wrote 'I will go home and take a HOT shower', instead of the proposed answer and was considered wrong. I see a big difference between taking a hot shower and a warm one, but then again, it might just be my english failing here