Klub je u biblioteci.

Because u means in/at here, and when it shows a static location, Serbian normally uses the locative case.

The basic form is biblioteka = library.
After u for location, it changes to biblioteci.

So:

• biblioteka = dictionary form
• u biblioteci = in the library

That is a normal spelling and sound change in Serbian.

The noun ends in -ka, and in forms like the locative singular, k often changes to c before i:

• biblioteka → biblioteci
• banka → banci

So this is not an irregular one-off form; it follows a common pattern.

Je is the 3rd person singular present form of biti = to be.

So it means is.

In Klub je u biblioteci, the structure is:

• Klub = subject
• je = is
• u biblioteci = location phrase

So literally: Club is in library.

Serbian does not have articles like English a/an and the.

So klub can mean:

• a club
• the club

And biblioteci can mean:

• in a library
• in the library

The exact meaning depends on context. English has to choose an article, but Serbian usually does not mark that directly.

Klub is in the nominative singular, because it is the subject of the sentence.

Its basic form is also klub, so there is no visible change here.

So:

• klub = nominative singular
• it is masculine
• it means club

Yes, Serbian word order is more flexible than English word order.

Klub je u biblioteci is a natural, neutral way to say it.

But other orders are possible, for example:

• U biblioteci je klub.

That could sound more like It is in the library that the club is or put more focus on the location.

So the basic meaning can stay the same, but the emphasis changes.

Serbian uses different prepositions depending on the type of place.

Here u is used because biblioteka is treated as an enclosed place or institution, so u biblioteci = in the library / at the library.

Compare:

• u kući = in the house
• u školi = at school / in school
• na stolu = on the table
• na fakultetu = at university

So with biblioteka, u is the normal choice.

Then Serbian usually uses the accusative, not the locative.

So:

• u biblioteci = in the library (location)
• u biblioteku = into/to the library (movement)

For example:

• Klub je u biblioteci. = The club is in the library.
• Idem u biblioteku. = I am going to the library.

This is a very important Serbian pattern:
location and movement toward often use different cases.

Yes. In standard Serbian, you normally keep the present-tense form of to be in a sentence like this.

So Klub je u biblioteci is correct.
Leaving out je would sound incomplete or ungrammatical in standard Serbian.

This is different from some other Slavic languages where the present-tense to be can be omitted.

A simple English-friendly approximation is:

kloob yeh oo bee-blee-oh-TEH-tsee

A few useful pronunciation points:

• j sounds like English y
• u sounds like oo
• c sounds like ts
• Serbian spelling is quite phonetic, so words are usually pronounced much as they are written

So:

• Klub ≈ kloob
• je ≈ yeh
• u ≈ oo
• biblioteci ≈ bee-blee-oh-TEH-tsee