Wir wünschen dir Glück.

Because dir is the dative form of du.

The verb wünschen often takes:

• a person in the dative: dir = to you
• a thing in the accusative: Glück = luck

So the structure is:

jemandem etwas wünschen
= to wish someone something

That is why German says:

• Wir wünschen dir Glück.

not

• Wir wünschen du Glück.

Glück is the accusative object of the sentence.

With wünschen, the thing being wished is usually in the accusative:

• Wir wünschen dir Glück.
• Ich wünsche dir Erfolg.
• Sie wünscht ihm alles Gute.

In this sentence:

• wir = subject
• dir = indirect object, dative
• Glück = direct object, accusative

Because Glück has no article here, you do not see a visible accusative ending.

German often uses abstract nouns like Glück, Geduld, Erfolg, Frieden, or Mut without an article when speaking in a general sense.

So:

• Wir wünschen dir Glück.
  means wishing luck in general.

You could compare:

• Wir wünschen dir Glück. = general luck
• Wir wünschen dir viel Glück. = a lot of luck, which is even more common in everyday speech

So the missing article is normal here.

The infinitive is wünschen = to wish.

Here it is conjugated for wir:

• ich wünsche
• du wünschst
• er/sie/es wünscht
• wir wünschen
• ihr wünscht
• sie/Sie wünschen

So Wir wünschen means we wish.

In German, pronouns often come earlier in the middle of the sentence than full nouns.

So dir naturally comes before Glück:

• Wir wünschen dir Glück.

This is the most normal order.

Because dir is a short pronoun, it usually appears before the noun object. A version like Wir wünschen Glück dir is not normal standard word order.

It is grammatical and understandable, but in everyday German, Wir wünschen dir viel Glück is more common.

Common versions are:

• Wir wünschen dir viel Glück.
• Ich wünsche dir viel Glück.
• Viel Glück!

So Wir wünschen dir Glück is fine, but it can sound a little more formal, literary, or simply less idiomatic than viel Glück.

Yes, Glück can mean both:

• luck
• happiness

In this sentence, it usually means luck, because with jemandem Glück wünschen, that is the normal meaning.

Examples:

• Viel Glück! = Good luck!
• Sie fanden ihr Glück. = They found their happiness.

So the context tells you which meaning is intended.

You would replace dir with Ihnen.

• informal singular: Wir wünschen dir Glück.
• formal singular/plural: Wir wünschen Ihnen Glück.

So if you are speaking politely to someone you do not know well, Ihnen is the right choice.

Yes. The only change is the subject:

• Ich wünsche dir Glück. = I wish you luck.
• Wir wünschen dir Glück. = We wish you luck.

The rest of the sentence stays the same because the object structure with dir and Glück does not change.

A rough English-friendly guide is:

VIR VUEN-shen deer GLUECK

A few important points:

• w in German sounds like English v
• ü is not an English sound; it is made with the tongue position of ee but with rounded lips
• sch sounds like English sh
• ch in Glück is the soft German ich/ach sound? Actually here Glück ends with ck, so it is just a hard k sound
• wir sounds like veer
• dir sounds roughly like deer
• Glück sounds roughly like glyook or glook with rounded lips, though neither spelling is perfect

If pronunciation is your focus, ü is the sound worth practicing most.

A very useful pattern is:

jemandem etwas wünschen

That means:

• wish someone something

Examples:

• Ich wünsche dir alles Gute.
• Wir wünschen euch viel Erfolg.
• Sie wünscht ihm Glück.

So the key lesson is:

• the person gets dative
• the thing gets accusative

That pattern will help you build many similar sentences correctly.