'To take a shower' is the standard American way to say it. If you want it to sound more informal, you can use the contraction 'gonna' instead of 'going to.' It's very common to not enunciate 'going to' and instead it turns into something sounding like 'gah-nuh.'
'I'm going to (gonna) take a shower.'
You can also say: 'I'm going to be (gonna be) in the shower,' it's like the prepositional case way of expressing this. Both ways are fine; maybe the second way is slightly better in this context because it emphasizes that you will be unavailable and in another place (i.e. in the shower and not at your desk) whereas the first is more emphasizing the activity.
As another example, if you couldn't do something in the context of being busy due to a vacation, you could say:
'Sorry, I can't, I'm gonna be on vacation.' In US English, we'd say it this way rather than 'I'll be vacationing.'