Nie wiem, w którym kierunku iść, więc patrzę na znak przy skrzyżowaniu.

Both can mean I don’t know in English, but they are used differently in Polish.

• nie wiem = I don’t know a fact, answer, direction, result, etc.
• nie znam = I don’t know / I’m not familiar with a person, place, book, language, etc.

Here the speaker does not know which way to go, so nie wiem is correct.

• Nie wiem, w którym kierunku iść. = I don’t know which direction to go.
• Nie znam tego miasta. = I don’t know this city.

Literally, it means in which direction.

Breakdown:

• w = in
• którym = which
• kierunku = direction

So the phrase asks about the direction within a set of possible directions: in which direction.

In natural English, we would often simply say which direction.

Because który has to change its form to match the noun and the case required by the preposition.

Here:

• kierunek is masculine
• after w in this meaning, Polish uses the locative case
• so który becomes którym
• and kierunek becomes kierunku

So:

• nominative: który kierunek = which direction
• locative: w którym kierunku = in which direction

This is a very common pattern in Polish: the question word changes form depending on case.

It is the locative singular form of kierunek.

The preposition w often requires the locative when it means in or inside a context/location.

So:

• basic form: kierunek
• after w: w kierunku or w którym kierunku

Even though English does not show this kind of change, Polish nouns regularly change form after prepositions.

Because after nie wiem Polish often uses an infinitive to express I don’t know what to do / where to go / how to start, etc.

So:

• Nie wiem, co zrobić. = I don’t know what to do.
• Nie wiem, jak zacząć. = I don’t know how to begin.
• Nie wiem, w którym kierunku iść. = I don’t know which direction to go.

The infinitive here works a lot like English to go, even though Polish often leaves out a separate word for to.

These verbs are related, but they are not interchangeable.

• iść = to go, to be going, on foot, in one direction
• chodzić = to go/walk habitually, repeatedly, or in various directions
• pójść = to go once, with a completed/future sense

In this sentence, the speaker is deciding which direction to take right now, so iść fits very well.

Compare:

• Nie wiem, w którym kierunku iść. = I don’t know which way to go.
• Nie wiem, gdzie chodzić po mieście. = I don’t know where to walk around in the city.
• Nie wiem, w którą stronę pójść. = I don’t know which way to go.
  This is also possible, but it sounds a bit more like making a single decision and setting off.

Więc means so, therefore, or thus.

It connects two clauses:

• Nie wiem... = I don’t know...
• więc patrzę... = so I look...

The comma is standard in Polish before conjunctions like więc when they join two full clauses.

So the structure is:

• I don’t know X, so I do Y.

Because the verb patrzeć normally goes with na when you say what you are looking at.

So:

• patrzeć na coś = to look at something

Examples:

• Patrzę na mapę. = I’m looking at the map.
• Patrzy na mnie. = He/She is looking at me.
• Patrzę na znak. = I’m looking at the sign.

Without na, the sentence would sound wrong in standard Polish.

It actually is in the accusative, but znak is an inanimate masculine noun, and for this type of noun the accusative singular looks the same as the nominative singular.

So:

• nominative: znak
• accusative: znak

That is why you see:

• na znak

Compare with a feminine noun:

• mapa → na mapę

So the case is changing, but the form happens to stay the same here.

It means by the intersection, near the intersection, or at the intersection.

Breakdown:

• przy = by, near, next to
• skrzyżowaniu = intersection, in the required case

So znak przy skrzyżowaniu is the sign by the intersection.

It suggests the sign is located near or at the intersection, not necessarily directly in the middle of it.

Because the preposition przy requires the locative case.

So:

• basic form: skrzyżowanie
• after przy: przy skrzyżowaniu

This is a fixed grammar pattern:

• przy domu = by the house
• przy stole = at the table
• przy skrzyżowaniu = by the intersection

Yes, but the meaning would shift a little.

• przy skrzyżowaniu = by / near the intersection
• na skrzyżowaniu = at the intersection, more directly located there

So:

• znak przy skrzyżowaniu sounds like a sign standing near the intersection
• znak na skrzyżowaniu sounds more like a sign located right at the intersection area

In many real situations, both may be possible, but przy often feels more natural for something standing beside the road.

Not completely. Polish word order is fairly flexible, but the original sentence is very natural.

Original:

• Nie wiem, w którym kierunku iść, więc patrzę na znak przy skrzyżowaniu.

You could rearrange parts for emphasis, for example:

• Patrzę więc na znak przy skrzyżowaniu, bo nie wiem, w którym kierunku iść.
• Na znak przy skrzyżowaniu patrzę, bo nie wiem, w którym kierunku iść.

But these alternatives change the emphasis or sound more marked. The original version is the most neutral and natural.

Yes, it is natural and correct. But Polish also has other common ways to express the same idea, for example:

• Nie wiem, w którą stronę iść.
• Nie wiem, gdzie iść.
• Nie wiem, którędy iść.

These are slightly different:

• w którym kierunku = in which direction
• w którą stronę = which way
• gdzie = where
• którędy = by which route / which way through

So the sentence given is perfectly good, just a bit more explicit and directional.