امروز در دانشگاه کلاس دارم و بعد از کلاس به کتابخانه میروم.

In Persian, the subject is often omitted because the verb ending already shows who is doing the action.

• دارم = I have
• می‌روم / میروم = I go / I am going

So من is not necessary here. You could add it for emphasis:

من امروز در دانشگاه کلاس دارم...

That would mean I have class today..., with extra emphasis on I.

It literally means I have class.

• کلاس = class
• دارم = I have

This is a very natural Persian way to talk about scheduled activities. So در دانشگاه کلاس دارم means I have class at the university.

Because Persian normally follows a subject-object-verb pattern, and the verb usually comes at the end of the clause.

So:

• در دانشگاه کلاس دارم = literally something like at the university class I have
• به کتابخانه می‌روم = literally to the library I go

This verb-final pattern is very common in Persian.

Persian does not use articles the same way English does.

In this sentence:

• دانشگاه can mean the university or a university, depending on context
• کلاس can mean class or a class
• کتابخانه can mean the library or a library

If Persian wants to make a/an clearer, it can use یک:

• یک کلاس دارم = I have a class

But very often, Persian simply leaves articles out and lets context do the work.

Yes, both can work.

• در دانشگاه = more neutral, standard, or slightly more formal
• تو دانشگاه = more conversational, everyday speech

Both mean in/at the university in this kind of sentence. The version with در sounds a bit more suitable for careful written Persian.

بعد از is a very common Persian expression meaning after.

• بعد = after / later
• از = from / of in this construction

Together, بعد از functions as a set phrase:

• بعد از کلاس = after class

You may also see بعدِ کلاس in Persian, which can mean the same thing, but بعد از کلاس is extremely common and very clear.

Because به is used to mark a destination with verbs of movement.

• به کتابخانه می‌روم = I go to the library

Here:

• به = to
• کتابخانه = library
• می‌روم = I go / I am going

So به is the word that shows where the speaker is going.

In standard modern spelling, yes: می‌روم is preferred.

The prefix می is usually written with a half-space before the verb:

• می‌روم
• می‌خوانم
• می‌بینم

But many people type it without the half-space:

• میروم

That form is still very commonly seen online or in informal typing, and it means the same thing.

Grammatically, it is a present form, but Persian often uses the present to talk about a planned or expected future action, especially when the time is already clear.

So in this sentence:

امروز ... بعد از کلاس به کتابخانه می‌روم

it can naturally mean:

• Today I go to the library after class
• Today I’m going to the library after class

Both are good English interpretations depending on context.

A helpful pronunciation guide is:

emruz dar daneshgâh kelâs dâram va ba'd az kelâs be ketâbkhâne mi-ravam

A few notes:

• امروز = emruz
• دانشگاه = dâneshgâh or daneshgâh
• کتابخانه = ketâbkhâne
• می‌روم in careful speech is mi-ravam
• In everyday spoken Persian, it is often said more like miram

So a more natural spoken version may sound closer to:

emruz tu dâneshgâh kelâs dâram o ba'd az kelâs be ketâbkhâne miram