Aĉetinte cepojn kaj terpomojn, ni revenas hejmen por kuiri.

Aĉetinte means having bought.

It is made from:

• aĉet- = the verb stem buy
• -int- = the active past participle marker
• -e is usually used for adverbs, but here the full ending is -inte, which is the standard adverbial participle ending for having done X

So:

• aĉeti = to buy
• aĉetinte = having bought

In this sentence, Aĉetinte cepojn kaj terpomojn means Having bought onions and potatoes or After buying onions and potatoes.

Because the sentence is using a shortened adverbial construction.

Compare:

• Aĉetinte cepojn kaj terpomojn, ni revenas hejmen por kuiri.
• Post kiam ni aĉetis cepojn kaj terpomojn, ni revenas hejmen por kuiri.

Both are valid. The first is more compact and elegant. It means that the buying happened before the returning home.

So aĉetinte is a neat way to say:

• after buying
• having bought

It is understood to be the same subject as in the main clause: ni = we.

So the sentence means:

• We, having bought onions and potatoes, return home to cook.

In Esperanto, with this kind of participial phrase, the implied subject is normally the same as the subject of the main verb. So this sentence does not naturally mean:

• After someone else bought onions and potatoes, we return home.

If you wanted a different subject, you would usually need to say it more explicitly.

Because they are the direct objects of aĉetinte / aĉeti.

• cepoj = onions
• terpomoj = potatoes
• cepojn / terpomojn = onions / potatoes as direct objects

In Esperanto, direct objects usually take -n.

So:

• Mi aĉetas cepojn. = I buy onions.
• Ni aĉetinte cepojn kaj terpomojn... = Having bought onions and potatoes...

The -j shows plural, and the -n shows direct object.

Because hejmen shows movement toward home.

This is a very common Esperanto pattern:

• hejme = at home
• hejmen = to home / homeward / going home

The -n here is not a direct object marker. It is the accusative of direction, used to show motion toward a place.

So:

• Ni estas hejme. = We are at home.
• Ni iras hejmen. = We are going home.

In your sentence:

• ni revenas hejmen = we return home

Revenas is the present tense form of reveni = to return / to come back.

So literally it means:

• we return
• we are returning
• sometimes in context, we come back

In Esperanto, tense is usually very straightforward:

• revenas = present
• revenis = past
• revenos = future

In an isolated teaching sentence, the present tense is often used for a general statement, a narrative present, or just as a simple example. If the whole story were clearly in the past, you might expect:

• Aĉetinte cepojn kaj terpomojn, ni revenis hejmen por kuiri.

Iri means to go in a general sense.

Reveni means to return / come back.

It is built from:

• re- = again / back
• veni = come

So reveni has the idea of coming back to a previous place.

In this sentence:

• ni revenas hejmen = we return home

This suggests that we were away from home and are now going back.

Por kuiri means to cook or more precisely in order to cook.

• por = for, in order to
• kuiri = to cook

So this part gives the purpose of returning home.

The whole idea is:

• We return home in order to cook.

This is a very common Esperanto pattern:

• Mi venis por helpi. = I came to help.
• Ŝi iris al la vendejo por aĉeti panon. = She went to the store to buy bread.

Yes. That is a very natural translation.

Aĉetinte often corresponds to:

• having bought
• after buying

Both work well in English. The best translation depends on style.

So these are all reasonable:

• Having bought onions and potatoes, we return home to cook.
• After buying onions and potatoes, we return home to cook.
• After we buy onions and potatoes, we return home to cook.
  This last one is possible in some contexts, but it is a little less exact than having bought, because aĉetinte clearly shows the buying is already completed.

Because Esperanto does not use la unless the nouns are definite or specifically identifiable from context.

So:

• cepojn kaj terpomojn = onions and potatoes, in a general sense
• la cepojn kaj la terpomojn = the onions and the potatoes, specific ones

English sometimes uses no article with plural nouns too, so this is fairly similar:

• We bought onions and potatoes.

If the speaker means some particular onions and potatoes already known to the listener, then la could be used.

Yes. Esperanto word order is quite flexible, as long as the meaning stays clear.

For example, these are possible:

• Aĉetinte cepojn kaj terpomojn, ni revenas hejmen por kuiri.
• Ni revenas hejmen por kuiri, aĉetinte cepojn kaj terpomojn.

The first version is more natural because it presents the earlier action first: first buying, then returning home.

So while word order can vary, the original order is the clearest and most natural here.

A few useful pronunciation points:

• ĉ sounds like ch in church
• j sounds like English y
• stress in Esperanto is always on the second-to-last syllable

So approximately:

• aĉetinte ≈ a-che-TIN-te
• cepojn ≈ TSE-poyn
  (with oj sounding roughly like oy)
• hejmen ≈ HAY-men

And:

• terpomojn ≈ ter-po-MOYN
• kuiri ≈ ku-EE-ri

The regular spelling and stress system is one of the nice things about Esperanto.