Posle škole mnogo učim na računaru.

Because posle requires the genitive case.

• škola = nominative singular, the basic dictionary form
• škole = genitive singular

So posle škole means after school.

This is a very common pattern in Serbian:

• posle ručka = after lunch
• posle časa = after class
• posle posla = after work

Yes, škole can mean two different things depending on context:

• genitive singular of škola = of school / after school
• nominative plural = schools

In posle škole, the preposition posle tells you that škole must be genitive singular, so the meaning is after school, not after schools.

Here mnogo means a lot or much.

It modifies the verb učim, so mnogo učim = I study a lot.

Compare:

• Mnogo učim. = I study a lot.
• Mnogo radim. = I work a lot.
• Mnogo čitam. = I read a lot.

Because Serbian usually does not need subject pronouns when the verb ending already shows who is doing the action.

• učim = I study / I am studying

The ending -m tells you it is first person singular: I.

So ja is optional:

• Učim. = I study.
• Ja učim. = I study.
  This version adds emphasis, as in I study.

Učim is the 1st person singular present tense of učiti.

• učiti = to study / to learn
• učim = I study / I learn

This verb can refer either to:

• studying in general
• learning something

In this sentence, because of the context posle škole and na računaru, it most naturally means I study.

It usually means a habitual action here.

So Posle škole mnogo učim na računaru. most naturally means:

• After school, I study a lot on the computer.

That is, this is something the speaker generally does.

Serbian present tense can also describe something happening right now, but in this sentence the meaning is more naturally habitual.

Because when talking about working, studying, writing, or doing something on a computer/device, Serbian commonly uses na.

So:

• na računaru = on the computer

This matches English fairly closely.

Also:

• na telefonu = on the phone
• na internetu = on the internet

Using u računaru would mean something more like inside the computer, which is not the meaning here.

Računaru is in the locative singular.

That is because na can take:

• accusative for direction/movement
• locative for location

Here there is no movement; it describes where the studying happens, so Serbian uses locative:

• na računaru = on the computer

From the noun:

• računar = computer
• računaru = on/in the computer context, locative singular after na

Yes. In this kind of sentence, na računaru often means both:

• on the computer
• using a computer

It is the normal Serbian way to express doing something by means of a computer.

So mnogo učim na računaru can naturally mean:

• I study a lot on the computer
• I do a lot of my studying using a computer

Yes. Serbian word order is fairly flexible.

The neutral order here is:

• Posle škole mnogo učim na računaru.

But you could also say:

• Mnogo učim na računaru posle škole.
• Na računaru mnogo učim posle škole.
• Posle škole na računaru mnogo učim.

These versions all mean roughly the same thing, but the emphasis changes a little. The original sentence sounds natural and neutral.

Yes. Nakon škole also means after school.

• posle škole = very common, everyday
• nakon škole = also correct, sometimes a bit more formal or stylistic depending on context

So both are fine:

• Posle škole mnogo učim na računaru.
• Nakon škole mnogo učim na računaru.

That depends on the regional standard.

• posle is standard in Ekavian forms
• poslije is standard in Ijekavian forms

So depending on the variety of Serbian or the broader regional standard being used, you may see either:

• Posle škole...
• Poslije škole...

Both mean the same thing.

Both are possible meanings of učiti, and context decides.

In this sentence, to study is the most natural translation:

• Posle škole mnogo učim na računaru. = After school, I study a lot on the computer.

But in other contexts, učiti can mean to learn:

• Učim srpski. = I am learning Serbian.
• Učim za test. = I am studying for a test.

So učiti covers both English ideas.

The letter č is pronounced like the ch in chocolate, but usually a bit firmer.

So:

• učim sounds roughly like OO-chim
• računaru sounds roughly like RA-chu-na-ru

A rough pronunciation of the full sentence is:

• PO-sle SHKO-le MNO-go OO-chim na ra-CHU-na-ru

That spelling is only approximate, but it gives the general idea.