Posle semafora skrećem desno, a moja prijateljica skreće levo.

Because posle requires the genitive case.

• semafor = traffic light
• posle semafora = after the traffic light

So:

• nominative: semafor
• genitive singular: semafora

This is a very common pattern in Serbian:

• posle škole = after school
• posle ručka = after lunch
• posle semafora = after the traffic light

Here it means after the traffic light or past the traffic light in the context of directions.

In route descriptions, Serbian often uses expressions like this to mark a point on the الطريق/route:

• posle mosta = after the bridge
• posle raskrsnice = after the intersection
• posle semafora = after the traffic light

So the idea is: once you pass the traffic light, the next action is turning.

These are different present-tense forms of the same verb.

The verb is skretati / skrenuti, meaning to turn.

Here you have:

• skrećem = I turn
• skreće = he/she/it turns

So the sentence contrasts two subjects:

• (ja) skrećem desno = I turn right
• moja prijateljica skreće levo = my friend turns left

The verb ending tells you the subject, which is why Serbian often does not need the pronoun ja.

Because Serbian usually omits subject pronouns when they are not needed.

The ending -em in skrećem already tells you it means I turn.

So:

• skrećem desno = I turn right
• ja skrećem desno = I turn right

Both are correct, but ja adds emphasis, something like:

• I’m the one turning right

In neutral speech, leaving out ja is more natural.

A often means and, but more specifically it often shows a contrast or comparison between two parts.

So here:

• Posle semafora skrećem desno, a moja prijateljica skreće levo.

means something like:

• After the traffic light, I turn right, while my friend turns left.

It is not a strong contradiction like but, but it does highlight a difference between the two actions.

Compare:

• i = and, simple addition
• a = and/while/but, with contrast

Here a is very natural because the speakers go in different directions.

Because here they are adverbs, not adjectives.

They describe how / in what direction someone turns:

• skrenuti desno = turn right
• skrenuti levo = turn left

Adverbs do not agree with nouns, so they stay the same.

Compare:

• desni put = the right-hand road / the right road (desni = adjective)
• skrećem desno = I turn right (desno = adverb)

So in this sentence, desno and levo are the correct forms.

Yes. desno / levo and udesno / ulevo can both be used in directional contexts.

For example:

• skrećem desno
• skrećem udesno

Both mean I turn right.

And:

• skreće levo
• skreće ulevo

Both mean she turns left.

The shorter forms desno and levo are very common and completely natural here.

Because Serbian, like English, often uses the present tense to describe:

• habitual actions
• route instructions
• a sequence in a story

So skrećem and skreće do not have to mean something happening literally right now. They can mean:

• I turn right
• my friend turns left

as a step in a route or a repeated action.

This is very normal in Serbian.

Because prijateljica means female friend and it is a feminine noun, so the possessive adjective must agree with it.

• moj = my (masculine)
• moja = my (feminine)
• moje = my (neuter)

So:

• moj prijatelj = my male friend
• moja prijateljica = my female friend

Agreement is important in Serbian: adjectives and possessives usually match the noun in gender, number, and case.

It specifically refers to a female friend.

• prijatelj = male friend
• prijateljica = female friend

In English, friend does not usually show gender, but Serbian often does.

So moja prijateljica clearly tells you the friend is female.

Yes, Serbian word order is fairly flexible, although some versions sound more natural than others depending on emphasis.

The given sentence:

• Posle semafora skrećem desno, a moja prijateljica skreće levo.

is a very natural neutral order.

You could also say:

• Skrećem desno posle semafora, a moja prijateljica skreće levo.

This still works, but it shifts the focus a bit.

Putting Posle semafora first is especially natural when giving directions, because it sets the location or point in the route before the action.

Yes: the letter ć is important.

• skrećem
• skreće

The sound ć is softer than English ch. English speakers often pronounce it too strongly. It is different from č, which is a harder sound.

Also note:

• c in semafora is not present there, but in Serbian in general c is pronounced like ts
• ć is a separate letter and sound

So it is useful to practice the difference between:

• c
• č
• ć

because they can distinguish words in Serbian.

Both exist, depending on the standard/dialect.

• In Ekavian Serbian: levo
• In Ijekavian forms used in some regions/standards: lijevo

Since your sentence has levo, it is using the common Ekavian Serbian form, which is standard in Serbia.

So for learners, desno and levo are perfectly normal standard Serbian forms.