Kaži mi gde je tvoj dom.

A word-by-word breakdown is:

• Kaži = say / tell
• mi = to me
• gde = where
• je = is
• tvoj = your
• dom = home

So the structure is literally: Tell to me where is your home.

Kaži is the imperative form of the verb kazati or closely related to reći, meaning to say or to tell.

Here, Kaži! means Tell! and it is:

• singular
• informal
• used when speaking to one person

So Kaži mi... means Tell me...

A more formal or plural version would be:

• Kažite mi gde je vaš dom. = Tell me where your home is.

Because mi is the dative form meaning to me, while mene is an accusative/genitive form meaning me in other kinds of constructions.

In Serbian, tell someone uses the person in the dative:

• Kaži mi = Tell me
• literally Say to me

So mi is exactly what you want after Kaži.

Because the most natural Serbian word order here is:

• gde je tvoj dom = where your home is / where is your home

The verb je often comes early in the clause. Serbian word order is more flexible than English, but this version is the standard, natural one.

Other word orders may be possible in special contexts for emphasis, but gde je tvoj dom is the normal choice.

Because dom is a masculine singular noun, and the possessive adjective must agree with it.

So:

• tvoj dom = your home for a masculine noun
• tvoja kuća = your house because kuća is feminine
• tvoje selo = your village because selo is neuter

This is grammatical agreement: Serbian adjectives and possessives change form to match the noun.

This is a very common question.

• dom usually means home in the emotional or personal sense
• kuća usually means house as a building

So:

• tvoj dom = your home
• tvoja kuća = your house

In some contexts, dom can also mean things like institution or residence, depending on the phrase, but here it clearly means home.

It is informal, because of Kaži and tvoj.

Informal singular:

• Kaži mi gde je tvoj dom.

Formal singular or plural:

• Kažite mi gde je vaš dom.

So if you are talking to a friend, child, or someone you address with ti, this sentence is fine.
If you are speaking politely or to more than one person, you would normally use Kažite and vaš.

A rough pronunciation guide for an English speaker:

• Kaži ≈ KAH-zhee
• mi ≈ mee
• gde ≈ gdeh or almost gdeh with both consonants pronounced
• je ≈ yeh
• tvoj ≈ tvoy
• dom ≈ dohm

A few useful notes:

• ž sounds like the s in measure
• j is like English y in yes
• gde may feel tricky because of the consonant cluster at the start

Yes, depending on the standard and regional variety.

• gde is common in Serbian
• gdje appears in some other South Slavic standards and regional speech

If you are learning standard Serbian, gde is the normal form to learn first.

Yes. Serbian often allows more than one natural phrasing. For example:

• Kaži mi gde ti je dom.
• Reci mi gde je tvoj dom.
• Reci mi gde ti je dom.

These all mean roughly the same thing.

A useful point:

• Reci mi is often extremely common in everyday speech
• ti je can add a more conversational feel, something like where your home is

So your original sentence is correct, but it is not the only natural way to say it.

Dom is in the nominative singular.

That is because it is the subject of the clause:

• gde je tvoj dom = where your home is

Here, dom is the thing that is somewhere, so nominative is the expected case.

Because Serbian does not need an extra word there. The clause simply starts with gde:

• Kaži mi gde je tvoj dom.

English also usually says Tell me where your home is, without that, so the structures are actually quite similar here.

Serbian often connects clauses directly, especially with question words like:

• gde = where
• kada = when
• zašto = why
• kako = how

Yes, it is grammatical and understandable, but in real conversation people might more often say something slightly more idiomatic, depending on context, such as:

• Odakle si? = Where are you from?
• Gde živiš? = Where do you live?
• Reci mi gde ti je dom. = Tell me where your home is.

So your sentence is correct, but whether it sounds fully natural depends on the situation and exactly what you want to ask.