Novo doba će početi danas.

Because doba is a neuter noun, and the adjective has to agree with it in gender, number, and case.

• novi = masculine singular
• nova = feminine singular
• novo = neuter singular

So:

• novi čovek = a new man
• nova knjiga = a new book
• novo doba = a new age / era

Here, both words are in the nominative singular.

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

The sentence structure is basically:

• Novo doba = subject
• će početi = will begin
• danas = today

So doba is the thing that will begin.

This is the normal way to form the future tense in Serbian.

It is made with:

• the clitic form of hteti (ću, ćeš, će, ćemo, ćete, će)
• plus the infinitive

So:

• ja ću početi = I will begin
• ti ćeš početi = you will begin
• ono će početi = it will begin

In your sentence, the subject is novo doba, which is grammatically third person singular, so the form is će početi.

Because Serbian often omits subject pronouns when they are not needed.

The noun novo doba already tells you what the subject is, so adding ono would be unnecessary:

• Novo doba će početi danas. = natural
• Ono će početi danas. = possible only if it is already clear from context
• Novo doba ono će početi danas. = unnatural in normal usage

So Serbian usually does not repeat the subject with a pronoun unless there is special emphasis or contrast.

Because će is a clitic, and Serbian clitics usually appear in the second position of the clause.

Here the first unit is Novo doba, so će comes right after it:

• Novo doba će početi danas.

This is very typical Serbian word order.

You may also see other arrangements for emphasis, for example:

• Danas će početi novo doba.
• Počeće danas novo doba. (more marked or literary)

But the given sentence is a very normal pattern.

Yes. That is also correct.

Serbian has another future form where the auxiliary is attached to the verb:

• počeće = počet(i) + će

So these are both correct:

• Novo doba će početi danas.
• Novo doba počeće danas.

The version with će + infinitive is very common and straightforward.
The version počeće can sound a bit more formal, compact, or literary in some contexts.

Početi is a perfective verb meaning to begin / to start.

That matters because in Serbian, perfective verbs focus on the action as a completed whole or as a single event. Here, the idea is the moment when the new age starts.

Its imperfective partner is počinjati:

• početi = to begin, to start (perfective)
• počinjati = to be beginning / to begin repeatedly or over time (imperfective)

In this sentence, početi is the natural choice because the sentence refers to the event of the new age starting.

Because Serbian has no articles.

English says:

• a new age
• the new age

Serbian simply says:

• novo doba

Whether it means a new age or the new age depends on context.

That is very normal in Serbian and other Slavic languages.

Yes. Serbian word order is fairly flexible, and moving danas changes the focus more than the core meaning.

For example:

• Novo doba će početi danas. = neutral
• Danas će početi novo doba. = emphasis on today
• Novo doba danas će početi. = possible, but less natural in everyday neutral speech

So the original sentence is a good neutral version.

Yes, doba is a common word, but it often refers to an age, era, period, or stage, while vreme is a more general word for time.

Compare:

• vreme = time in general
• doba = age / era / period

So novo doba sounds like a new age / a new era, not just new time in a general sense.

That is why doba fits well here.

A rough guide is:

• će ≈ cheh (with a softer ch sound)
• početi ≈ po-CHE-ti

A few notes:

• č is like ch in church
• ć is also a kind of ch sound, but softer than č
• Serbian spelling is very phonetic, so words are usually pronounced much as they are written

For an English speaker, the hardest part is usually hearing and producing the difference between č and ć.

Sometimes Serbian can use the present tense for future meaning, especially with time expressions, but not every verb works the same way, and this sentence is most naturally expressed with the future:

• Novo doba će početi danas.

That is the clear, standard way to say A new age will begin today.

Using the future here is the safest and most natural choice for learners.