Trebalo bi da danas odem u poštu, jer moram da pošaljem paket.

It means should or ought to.

So Trebalo bi da danas odem u poštu means something like I should go to the post office today.

It sounds softer than a direct obligation. In the same sentence, moram da pošaljem paket is stronger: I have to send a package. So the logic is:

• trebalo bi = recommendation / sensible thing to do
• moram = real necessity

That is why the sentence feels natural:
I should go to the post office today, because I have to send a package.

Both are possible, but they work a little differently.

• Trebalo bi da odem = an impersonal, very common way to say I should go
• Trebao bih da odem = a more personal form, also meaning I should go

With trebao bih / trebala bih, the form agrees with the speaker’s gender:

• trebao bih = if the speaker is male
• trebala bih = if the speaker is female

But trebalo bi stays neutral and is extremely common in everyday Serbian.

So:

• Trebalo bi da danas odem u poštu.
• Trebao bih da danas odem u poštu.
• Trebala bih da danas odem u poštu.

All can mean basically the same thing.

Because Serbian very often uses da + present-tense verb where English uses an infinitive.

English:

• I should go
• I have to send

Serbian:

• trebalo bi da odem
• moram da pošaljem

This da + verb pattern is one of the most important things for English speakers to get used to in Serbian.

So instead of an English-style infinitive like to go, Serbian usually says:

• da idem / da odem
• da šaljem / da pošaljem

This is about aspect.

• ići / idem = imperfective
• otići / odem = perfective

Here the speaker is talking about one complete trip to the post office, as a single finished action. That is why odem is natural.

Compare:

• Trebalo bi da danas odem u poštu.
  = I should go to the post office today.
  (one trip, completed event)

• Trebalo bi da idem u poštu češće.
  = I should go to the post office more often.
  (habitual/repeated idea)

So odem fits because the sentence is about one specific errand.

That is normal in Serbian.

Odem is formally a present-tense form of the perfective verb otići, but perfective present forms often refer to a future or intended completed action, especially after da.

So in:

• trebalo bi da odem
• moram da pošaljem

the verbs are present in form, but the meaning is about something that still needs to happen.

This is very common and natural.

Because u with the accusative is used for motion toward a place.

Here the idea is go into / go to the post office, so:

• u poštu

Compare:

• Idem u poštu. = I’m going to the post office.
• Ja sam u pošti. = I am in the post office.

So:

• u + accusative = movement toward
• u + locative = location inside

Because it is in the accusative singular.

The dictionary form is:

• pošta = post office / mail

After u when there is movement, Serbian uses the accusative:

• u poštu

This is a normal feminine noun pattern:

• nominative: pošta
• accusative: poštu
• locative: pošti

So in this sentence:

• odem u poštu = I go to the post office
• sam u pošti = I am in the post office

Yes, in many situations you may hear ići na poštu as well.

Very roughly:

• u poštu can suggest going into the post office building
• na poštu can sound more like going to the post office as a destination/institution

In everyday speech, both can occur, depending on region and habit. But in this sentence, u poštu is completely natural and easy to understand.

Again, this is aspect.

• slati / šaljem = imperfective
• poslati / pošaljem = perfective

The sentence is about sending one package, as one completed action. That is why perfective pošaljem is used.

Compare:

• Moram da pošaljem paket.
  = I have to send a package.
  (one completed sending)

• Često šaljem pakete.
  = I often send packages.
  (habitual action)

So pošaljem fits the idea of a single task that needs to be completed.

It is changing by function, but the form happens to stay the same.

Paket is a masculine inanimate noun. In the singular, masculine inanimate nouns often have the same form in:

• nominative: paket
• accusative: paket

Here it is the direct object of pošaljem, so it is accusative:

• pošaljem paket

But it looks identical to the dictionary form.

Yes, Serbian word order is fairly flexible.

In the sentence:

• Trebalo bi da danas odem u poštu...

putting danas before odem gives a natural emphasis to today.

You could also say:

• Trebalo bi da odem danas u poštu.
• Danas bi trebalo da odem u poštu.

These all mean roughly the same thing, but the emphasis shifts slightly.

The original version sounds very normal and balanced.

Because jer means because and introduces a subordinate clause.

So the sentence has two parts:

• Trebalo bi da danas odem u poštu
• jer moram da pošaljem paket

In standard Serbian writing, a comma is normally used before jer.

Yes. It shows two different levels of obligation.

• Trebalo bi da odem u poštu = going to the post office is what I should do
• Moram da pošaljem paket = sending the package is what I must do

So the second clause explains the first one:

I should go to the post office today, because I have to send a package.

This is a very natural combination in Serbian.