Posle te promene naš stan će biti noviji i lepši.

Because posle means after and it requires the genitive case.

So:

• ta promena = that change (nominative)
• te promene = of/after that change (genitive)

In this sentence, posle te promene means after that change.

It is genitive singular.

The base form is:

• promena = change

After posle, it becomes:

• promene

This is a very common pattern in Serbian: many prepositions require a specific case, and posle takes the genitive.

Because te has to agree with promene in gender, number, and case.

The noun promena is:

• feminine
• singular
• here in the genitive

So the demonstrative must match it:

• ta promena = that change
• te promene = of/after that change

Posle means after.

Yes, you can also say nakon:

• Posle te promene...
• Nakon te promene...

Both are correct.
Very roughly:

• posle is very common and everyday
• nakon can sound a bit more formal or bookish in some contexts

But in this sentence, both work well.

This is the future tense in Serbian.

• će = the future auxiliary
• biti = to be

So će biti means will be.

The structure is:

• naš stan će biti... = our apartment will be...

This is one of the standard ways Serbian forms the future.

Yes, very often you can.

So these are both possible:

• Naš stan će biti noviji i lepši.
• Naš stan biće noviji i lepši.

However, će biti is the more neutral and common form in this kind of sentence for many learners to start with. You will also see biće a lot in real usage.

Because the sentence is comparing the apartment after the change with how it is now/before.

So:

• nov = new
• noviji = newer

and

• lep = nice / pretty / attractive
• lepši = nicer / prettier

The idea is:

After that change, our apartment will be newer and nicer.

Even if English might sometimes sound a little unusual with newer, grammatically the Serbian is clearly making a comparison.

Because Serbian comparatives are not all formed with one single ending.

For example:

• nov → noviji
• lep → lepši

Both mean more X, but the form depends on the adjective.

This is something learners usually need to get used to: comparative forms are partly patterned, but not always fully predictable from English.

They agree with stan.

Here, stan is:

• masculine
• singular
• nominative

So the comparative adjectives describing it also appear in the matching form:

• noviji
• lepši

This is because after biti (to be), Serbian normally uses predicate adjectives that agree with the subject.

Because the normal, neutral order is for a possessive adjective to come before the noun:

• naš stan = our apartment

You can sometimes put the possessive after the noun, but that usually sounds marked, poetic, or emphatic. For everyday Serbian, naš stan is the standard order.

Because stan is the subject of the sentence.

The subject is normally in the nominative case, and here the basic form is:

• stan = apartment / flat

So:

• naš stan će biti... = our apartment will be...

In this sentence, lepši most naturally means nicer or more attractive.

For a home or apartment, lep often means something like:

• nice
• pretty
• attractive
• pleasant-looking

So lepši here is probably best understood as nicer rather than something dramatic like more beautiful.

Because in a neutral Serbian sentence with an expressed subject, the usual pattern is:

• subject + će + infinitive

So:

• naš stan će biti

Serbian word order is fairly flexible, but this is a very natural and standard arrangement. Clitic words like će have their own placement rules, and learners usually first meet them in patterns like this one.