Posiedźmy tutaj.

It is the 1st person plural imperative of posiedzieć.

In other words, it is the Polish let's ... form:

• posiedźmy = let's sit for a while

This form includes the speaker and at least one other person.

In this verb, po- adds the idea of doing something for a limited time or for a while.

So:

• siedzieć = to be sitting
• posiedzieć = to sit for a while

That is why Posiedźmy tutaj often feels closer to:

• Let's sit here for a bit than just the more neutral
• Let's sit here

Both are possible, but they are not exactly the same.

• siedźmy tutaj = let's be sitting here / let's stay sitting here
• posiedźmy tutaj = let's sit here for a while

The version with po- sounds more like a suggestion to spend some time sitting there. It often feels more natural when you mean let's rest here a bit or let's stay here seated for a little while.

This is a very useful distinction:

• usiądźmy tutaj = let's sit down here
• posiedźmy tutaj = let's sit here for a while

So:

• usiądźmy focuses on the movement into a sitting position
• posiedźmy focuses on spending some time sitting

If you are still standing and want to emphasize actually sitting down, Usiądźmy tutaj is often the better choice.
If the main idea is let's stay here seated for a bit, then Posiedźmy tutaj fits well.

Here, -my is part of the 1st person plural imperative ending, the Polish equivalent of let's ...

Compare:

• zróbmy = let's do
• chodźmy = let's go / come on
• usiądźmy = let's sit down
• posiedźmy = let's sit for a while

So when you see this kind of form, a good first guess is often let's ...

Because Polish imperatives often use a special stem, not just the infinitive minus the ending.

So you get:

• siedzieć → siedź! = sit!
• siedźcie! = sit! (plural / you all)
• siedźmy! = let's sit!

And the prefixed verb behaves the same way:

• posiedzieć → posiedź!
• posiedźcie!
• posiedźmy!

This is best learned as a pattern rather than trying to build it mechanically from the infinitive every time.

tutaj means here.
Yes, it is very close to tu, which also means here.

In many situations, they are interchangeable:

• Posiedźmy tutaj.
• Posiedźmy tu.

A rough difference:

• tu = shorter, very common, a bit more conversational
• tutaj = slightly fuller, sometimes a little more emphatic or explicit

But the difference is small, and both are normal.

Yes. Polish word order is fairly flexible.

The neutral order here is:

• Posiedźmy tutaj.

You could also say:

• Tutaj posiedźmy.

But that sounds more marked, as if you are emphasizing here specifically, maybe in contrast to somewhere else:

• Not there — let's sit here.

So the original order is the most natural default.

No. tutaj is an adverb, not a noun, so it does not change for case.

That is helpful to remember:

• tutaj = here
• it stays tutaj regardless of the rest of the sentence

A careful pronunciation is approximately:

[pɔˈɕɛd͡ʑmɨ ˈtutaj]

A rough English-friendly guide could be:

• po-SHYEDJ-mih TOO-tie

A few important points:

• si here sounds like a soft shy / ś sound
• dź is a very soft sound somewhat like j in jeans, but more palatal
• y in my is the Polish y, which does not have an exact English equivalent
• tutaj is stressed on the first syllable: TU-taj

By itself, it usually sounds like a friendly suggestion.

The let's form in Polish is not automatically rude. Its tone depends a lot on:

• context
• intonation
• relationship between speakers

So Posiedźmy tutaj. is normally just a natural suggestion like:

• Let's sit here for a while.

With an exclamation mark, it can sound more energetic:

• Posiedźmy tutaj!

With a full stop, it can sound calmer and more tentative:

• Posiedźmy tutaj.