بعد العمل اذهب الى الجامعة، ثم اقرا صحيفة في المكتبة.

Arabic often puts time expressions at the beginning of a sentence to set the scene first. So بعد العمل means after work, and placing it first is very natural.

You could also say اذهب إلى الجامعة بعد العمل. That would also be correct. Starting with بعد العمل just gives the time frame right away.

بعد means after, but grammatically it is treated like a noun of time, not just a simple preposition. It is followed by a noun in a genitive relationship.

So:

• بعدَ = after
• العملِ = the work / work

Together they form a kind of construct phrase: after work.

In fully vowelled Arabic, this would be بعدَ العملِ.

In Arabic, the verb itself often tells you who the subject is. Since اذهب and اقرأ are imperative forms, they already mean you go and you read.

So Arabic does not need a separate subject pronoun here.

If you added أنت, it would usually sound more emphatic, like you, go...

These forms are addressed to one male person: second person masculine singular.

So:

• اذهب = go! said to one male
• اقرأ = read! said to one male

If you were speaking to one female, you would say:

• اذهبي
• اقرئي

This is a very common thing for English speakers to notice, because English imperative forms do not change for gender.

Because the two prepositions express different ideas:

• إلى = to, toward
• في = in, inside

So:

• اذهب إلى الجامعة = go to the university
• في المكتبة = in the library

Arabic is being very precise here: first there is movement toward a place, then an action happening inside a place.

ثم means then, and it shows sequence: one action happens, and after that, the next action happens.

So:

• ثم = then, after that
• و = and

If you used و, the sentence would simply link the two actions. ثم is better if you want the meaning of first do this, then do that.

Because الـ is the definite article, meaning the.

So:

• الجامعة = the university
• المكتبة = the library
• صحيفة = a newspaper / newspaper

Here, the sentence refers to a specific university and a specific library, but not necessarily a specific newspaper. That is why صحيفة is indefinite.

If it meant read the newspaper, it would be اقرأ الصحيفة.

Yes. In all three words, the ل of الـ is pronounced.

That is because the first letters of these words are moon letters, not sun letters:

• العمل starts with ع
• الجامعة starts with ج
• المكتبة starts with م

So you say:

• al-ʿamal
• al-jāmiʿa
• al-maktaba

The l sound stays clear.

In isolation, اقرأ is pronounced roughly iqraʾ.

Important points:

• the first vowel is a short i
• ق is a deep q sound, not an English k
• the last sound is hamza: a glottal stop, like a brief catch in the throat

So it is not just iqra. The final hamza matters: iqraʾ.

In careful Modern Standard Arabic, the more standard spellings are:

• إلى
• اقرأ

A useful detail:

• إلى needs the initial hamza under the alif
• اقرأ needs the final hamza on the line
• اذهب is already fine as written, because its initial alif is a hamzat al-wasl, which is normally written as a plain alif

In everyday typing, some people omit hamza signs, but in formal writing learners should aim for إلى and اقرأ.

A fully vowelled version would be:

بَعْدَ العَمَلِ، اِذْهَبْ إِلَى الجَامِعَةِ، ثُمَّ اِقْرَأْ صَحِيفَةً فِي المَكْتَبَةِ.

A few useful things to notice:

• بعدَ has fatḥa because it is functioning as an adverb of time
• العملِ is genitive after بعد
• اِذْهَبْ ends in sukūn because it is an imperative
• الجامعةِ is genitive after إلى
• اِقْرَأْ is also an imperative
• صحيفةً is the direct object, so it is accusative
• المكتبةِ is genitive after في

In normal Arabic texts, these short vowels are usually not written.

Yes. That is also correct and natural.

Arabic has some flexibility with word order, especially with time expressions like بعد العمل. The difference is mostly one of emphasis:

• بعد العمل اذهب إلى الجامعة = After work, go to the university
• اذهب إلى الجامعة بعد العمل = Go to the university after work

Both are good. The first one highlights the time first.

Because everyday Arabic writing usually leaves out short vowel endings.

In a fully vowelled sentence, it would be صحيفةً, with tanwīn fatḥ because it is an indefinite direct object.

But in normal unvowelled writing, that appears simply as صحيفة.

This is very common in Arabic: learners are expected to supply the case ending from grammar and context, even when it is not written.