I go take a shower first. vs I take a shower fist. What's the difference?

It really depends on context. In the right context, neither is wrong. You can [go and take] a shower, or you can [take] a shower, or you can [go] shower. The action of travelling to the shower is always implied, but using “go” gives a bit more emphasis on the action. All of these below are correct: - “When I get home, I go take a shower first, then I make dinner” - “I take a shower first, then I make dinner” - “I’ll go take a shower first, then we can start dinner” - “I’ll go shower before we eat”

"i go take a shower" is very poor grammar generally. it's used but is lazy/ americanism "I go take a shower first" means i'll have a shower before i do anything "i go and take a shower fist" doesn't mean anything "I'll take a shower " or "i'll go and take a shower" is better

There’s not a meaningful difference between “take a shower” and “go take a shower” Either way the idea of taking a shower includes going to the place where you’re going to be showering.

A shower fist would be like you punch someone with a wet hand. 

So no. You wouldn’t have to specify where you are taking the shower because the majority of people will understand implicitly that if you’re taking a shower, you’re probably doing it in the bathroom. Saying “I’m going to take a shower in the bathroom” would be redundant, and kind of a weird thing to say. You are also missing a verb imo. I think the best syntax to get your meaning across would be “I will take a shower first” which means there is something you are going to do in the future, but before you do that, you want/need to shower. I wouldn’t say it’s a phrase people say, but similar idiomatic expressions might be “jump in the shower” “hop in the shower”. These phrases also imply speed - that you won’t be in the shower for very long. Hope all this is helpful!

Neither of these look like well formed sentences to me.

I'll go take a shower first would be future tense. "I take a shower first" would be past, like describing your own behavior. "I go take a shower first" , would also be past , would be odd and would still be describing your own behavior. If you're describing steps to get ready then yes, the second one adds moving to the room from another room. Still basically an expression.

Fisting is a different thing entirely.

I've heard expressions like ”go take a shower”. I'm not sure if adding ”go” changes the meaning much, it's just an expression. I assume it's short for ”go and'. ”Go take a hike” is a common way of telling someone to go away. Another place ”go” is used is the common insult ”Go and get f---ed”. I guess it just adds some emphasis. Speaking as an Australian.

You wouldn't say either, really. It'd be, "I'll (I will) go take a shower first." Or, "I am going to take a shower first." Both mean the same thing, they're both in the near future. The use of go makes it seem like you're going to be doing it now or very close to now. If you want to say it like you're doing it right now (like you're in the bathroom and the water is on), then it'd be "I'm (I am) taking a shower first." (Present tense) caveat is, this can also be used for near present as well. This explanation makes me grateful that I'm an English native. 🤣 Good luck with your learning!

I'd use 'have' rather than 'take'. Clearly you need to 'go' to the room with the shower, so 'go' is extraneous. I'll have a shower first.

I make a fist in the shower but I don't tell my wife what I am doing

both are wrong

You wouldn’t really say “I go take a shower first.” That starting phrase “I go…” would be for describing something you do but are not currently doing.” For example, I could say, “I go to the store on weekends” or “I go to Reddit when I need advice.”

Haven’t seen someone say this yet, but I think saying “Let me take a shower first” would be the best thing to say here

If someone is asking about your morning routine, as in, "What do you do when you wake up in the morning and get ready for the day?" you could use either "I go take a shower first," or "I take a shower first." Yes, using "go" implies that you're leaving your bedroom to go to wherever you take your shower, but no one would think you took a shower in your bedroom if you left out "go."

In my opinion, "go" here is superfluous. It's like saying "Stand up and walk to the shops." I mean, you could just say "Walk to the shops", which, unless your interlocutor is severely logically challenged, involves standing up. So when people say "I'll go and x", it's entirely to make the sentence sound less abrupt when speaking. In writing, it has no function.

If I'm explaining why I'm getting up and leaving the room, I might say, "I'm gonna go take a shower." But "I go take a shower" isn't quite right.

Your idea that destination somehow plays a part in the one version over the other is not correct. Both versions look like clumsy translations from an asian language, where the translator (or translation software) struggles with verb tenses and typically uses "first" where the correct word in English would be "now". A native English speaker would choose a different way of expressing it.