Ja danas idem u školu brže nego moja sestra.

That is true: Serbian usually does not need the subject pronoun, because the verb ending already tells you who is doing the action.

• idem = I go / I am going

So a very neutral version would simply be:

• Danas idem u školu brže nego moja sestra.

Adding ja gives extra emphasis or contrast, for example:

• I am going faster, not someone else.
• I am the one going today.

So ja is grammatically optional here.

Serbian does not have a separate continuous form like English I am going. The present tense idem can cover several English meanings depending on context:

• I go
• I am going
• sometimes even I’m going today / I’m going later

Because the sentence has danas (today), English usually prefers I am going or I’m going today, while Serbian still just uses the normal present tense idem.

Idem is the 1st person singular present tense of ići (to go).

A few present-tense forms of ići are:

• idem = I go / am going
• ideš = you go
• ide = he/she/it goes
• idemo = we go
• idete = you go
• idu = they go

So idem clearly shows that the subject is I, which is another reason ja can be omitted.

Because u changes meaning depending on whether there is movement toward a place or location inside a place.

Here, idem u školu means I am going to school, so it shows movement toward a destination. After u in that meaning, Serbian uses the accusative case.

• dictionary form: škola
• accusative singular: školu

So:

• u školu = to school

Not:

• u škola — incorrect
• u školi — that would mean in school / at school

This is a very important Serbian pattern:

• u školu = to school → motion/direction → accusative
• u školi = in school / at school → location → locative

Compare:

• Idem u školu. = I’m going to school.
• Ja sam u školi. = I am at school / in school.

English uses different prepositions or phrasing, while Serbian often changes the case.

Here brže is an adverb in the comparative, meaning faster.

It comes from:

• brzo = quickly, fast
• brže = faster

In this sentence it describes how the action of going happens:

• idem brže = I go faster / I’m going faster

So it is not describing a noun like a faster car would. It is describing the verb idem.

Here nego means than in a comparison.

• brže nego moja sestra = faster than my sister

Yes, Serbian can also often use od for comparison:

• brže od moje sestre

Both can be natural, but there is a small structural difference:

• nego often introduces a comparison that feels like a shortened clause
• od is followed by the genitive case

So:

• nego moja sestra → moja sestra stays nominative
• od moje sestre → moje sestre is genitive

Because after nego, Serbian often leaves out a repeated verb, and the listener understands it.

So:

• Ja danas idem u školu brže nego moja sestra.

really means something like:

• Ja danas idem u školu brže nego moja sestra ide.
• I am going to school faster than my sister is.

Since moja sestra is the subject of the implied verb ide, it stays in the nominative case.

If you used od instead, then you would need the genitive:

• brže od moje sestre

Strictly speaking, it is comparing the action or manner of going.

The full sense is:

• I am going to school faster than my sister goes.

So the comparison is really between:

• my speed of going
• my sister’s speed of going

That is why the omitted verb idea is helpful. Serbian often leaves that repeated verb unsaid when the meaning is obvious.

Serbian word order is fairly flexible, though some versions sound more neutral than others.

These are all possible, with slightly different emphasis:

• Ja danas idem u školu brže nego moja sestra.
• Danas idem u školu brže nego moja sestra.
• U školu danas idem brže nego moja sestra. — more marked
• Brže idem u školu danas nego moja sestra. — possible, but less neutral

The most neutral choices are usually:

• Danas idem u školu brže nego moja sestra.
• Ja danas idem u školu brže nego moja sestra. if you want to emphasize I

So yes, you can often move things around, but the focus changes.

Usually it means something like today I’m going to school or I am going to school today. It does not have to mean that the person is literally walking there at this exact moment.

Because Serbian present tense is broader than English, context decides the exact meaning. Depending on the situation, danas idem u školu could mean:

• I’m going to school today.
• I am going to school now, today.
• Today I go to school.

In natural English, I’m going to school today is often the best match.

Yes, it is understandable and grammatical. But in everyday speech, a native speaker might choose a slightly more neutral version such as:

• Danas idem u školu brže nego moja sestra.
• Danas idem u školu brže od moje sestre.

The version with ja is especially likely when the speaker wants contrast or emphasis.

So the sentence is fine; just remember that Serbian often drops ja unless there is a reason to stress it.