Ja sam stavila ključ u torbu.

A word-by-word breakdown is:

• Ja = I
• sam = am, but here it works as the auxiliary verb used to form the past tense
• stavila = put / placed (feminine singular past participle)
• ključ = key
• u = in / into
• torbu = bag in the accusative case

So the structure is literally something like I am put key into bag, but the real English meaning is I put the key in/into the bag.

Yes, Serbian often drops subject pronouns because the verb form already tells you who the subject is.

So:

• Stavila sam ključ u torbu = I put the key in the bag
• Ja sam stavila ključ u torbu also means the same thing, but ja adds emphasis

Using ja can suggest:

• contrast: I put it in the bag, not someone else
• clarity: you want to make the subject extra explicit
• natural speech rhythm in a particular context

So ja is not required, but it is perfectly correct.

This is the past tense in Serbian, often called the perfect.

It is formed with:

• the present tense of biti = to be
• plus the past participle

Here:

• sam = first person singular of biti
• stavila = past participle of staviti

So:

• ja sam stavila = I put / I have put

In everyday Serbian, this is the normal way to talk about completed past actions.

Because the speaker is female.

In Serbian past tense, the participle agrees with the gender and number of the subject:

• stavio = masculine singular
• stavila = feminine singular
• stavilo = neuter singular
• stavili / stavile = plural

So:

• a woman says Ja sam stavila ključ u torbu
• a man says Ja sam stavio ključ u torbu

This is a very important feature of Serbian past tense.

The infinitive is staviti, meaning to put or to place.

So the forms are related like this:

• staviti = to put
• stavila = put/placed, said by a female speaker
• stavio = put/placed, said by a male speaker

Staviti is also a perfective verb, which means it usually refers to a completed action.

It does take the accusative case, but for many masculine inanimate nouns in Serbian, the accusative singular is the same as the nominative singular.

So:

• nominative: ključ
• accusative: ključ

Because ključ is masculine and inanimate, its form stays the same here.

That is why:

• Imam ključ = I have a key
• Stavila sam ključ u torbu = I put the key into the bag

The case is still accusative, even though the form does not visibly change.

Because torba is the object of the preposition u here in the sense of movement into something, so it takes the accusative case.

The noun changes like this:

• nominative: torba = bag
• accusative singular: torbu

So:

• torba = the basic dictionary form
• u torbu = into the bag

This is one of the most common case patterns in Serbian.

Because in Serbian, u can go with different cases depending on whether there is:

• movement toward a place → accusative
• location in a place → locative

Here, the action is moving the key into the bag, so Serbian uses:

• u + accusative → u torbu

Compare:

• Stavila sam ključ u torbu = I put the key into the bag
• Ključ je u torbi = The key is in the bag

Notice the difference:

• u torbu = direction, movement
• u torbi = location

Yes, Serbian word order is fairly flexible.

You can also say:

• Stavila sam ključ u torbu
• Ključ sam stavila u torbu
• Ja sam ključ stavila u torbu

The basic meaning stays the same, but the emphasis changes.

A useful thing to know is that sam is a clitic, and clitics usually appear near the beginning of the sentence, often in the second position.

That is why forms like these are natural:

• Ja sam stavila...
• Ključ sam stavila...
• Juče sam stavila...

But sam usually does not stand completely freely wherever you want.

Often, yes.

In many neutral situations, Serbian prefers to leave out the subject pronoun, so:

• Stavila sam ključ u torbu sounds very natural and neutral

Adding ja makes the subject more explicit or emphatic:

• Ja sam stavila ključ u torbu can sound like I put the key in the bag as opposed to someone else

So both are correct, but they do not always sound equally neutral in context.

This is a question of aspect.

• staviti → perfective
• stavljati → imperfective

So:

• stavila sam ključ u torbu suggests a completed action: I put the key into the bag
• stavljala sam ključ u torbu would usually suggest repeated action, ongoing action, or background context, depending on the sentence

For example:

• Stavila sam ključ u torbu = I put the key in the bag
• Stavljala sam stvari u torbu = I was putting things into the bag / I used to put things into the bag

In your sentence, stavila is the normal choice for one completed act of putting the key in the bag.