По дороге домой я забегу в аптеку.

Literally, it is something like along the road home or on the way home.

• по дороге = on the way / along the road
• домой = homeward / to home

As a whole, по дороге домой is a very natural Russian way to say on the way home.

Because the preposition по here requires the dative case.

• dictionary form: дорога = road, way
• dative singular: дороге

So:

• по дороге = along the road / on the way

This is a very common pattern in Russian:

• по пути = on the way
• по улице = along the street
• по дороге домой = on the way home

Because домой expresses motion toward home, while дома means at home.

Compare:

• Я иду домой. = I’m going home.
• Я дома. = I’m at home.

In your sentence, the person is still moving and the phrase means on the way home, so домой is the correct choice.

Here забегу means I’ll pop in, I’ll drop by, or I’ll run in briefly.

It comes from the verb забежать. Literally, that verb is related to running in, but in everyday speech it often means making a quick stop somewhere.

So:

• забежать в аптеку = to pop into the pharmacy / to drop by the pharmacy

It suggests a short, quick visit, not a long stay.

Because забегу is a perfective verb form.

In Russian, perfective verbs do not normally have a true present tense. Their “present-looking” forms actually refer to the future.

So:

• забегу = I will drop by
• not I drop by in the present

This is the 1st person singular form:

• я забегу = I will drop by

The prefix за- often adds the idea of going in somewhere briefly or making a quick stop.

Compare the basic idea:

• бежать = to run
• забежать = to run in / stop by briefly

In this sentence, it does not necessarily mean literal running. It just gives the sense of a quick visit.

That is why забегу в аптеку sounds like:

• I’ll pop into the pharmacy
• I’ll stop by the pharmacy quickly

Because the sentence describes movement into the pharmacy, so Russian uses в + accusative.

• аптека = pharmacy
• accusative singular: аптеку

Compare:

• Я забегу в аптеку. = I’ll go into the pharmacy.
• Я в аптеке. = I’m in the pharmacy.

So:

• в аптеку = into the pharmacy
• в аптеке = in the pharmacy

Because в аптеку means going inside the pharmacy, while к аптеке means going toward / up to the pharmacy building.

Compare:

• Я зайду в аптеку. = I’ll go into the pharmacy.
• Я подойду к аптеке. = I’ll walk up to the pharmacy.

Since the meaning here is to stop in at the pharmacy, в аптеку is the natural choice.

Yes, very often it can.

Russian usually allows the subject pronoun to be omitted when the verb ending already makes the subject clear.

So both are possible:

• По дороге домой я забегу в аптеку.
• По дороге домой забегу в аптеку.

Both mean the same thing. Including я can add a little emphasis or clarity, but it is not always necessary.

Yes, Russian word order is flexible.

This version puts по дороге домой first, which sets the scene: on the way home.

Other orders are possible, for example:

• Я по дороге домой забегу в аптеку.
• В аптеку я забегу по дороге домой.

These all express roughly the same basic meaning, but the emphasis changes:

• По дороге домой... emphasizes when
• В аптеку... emphasizes where
• Я... emphasizes who

The original order is very natural and neutral.

Usually yes. Аптека is the normal word for a pharmacy or drugstore.

Depending on context, English might translate it as:

• pharmacy
• drugstore
• chemist’s (in British English)

In most learning contexts, pharmacy is the safest translation.

Yes. По пути домой also means on the way home and is very common.

Compare:

• по дороге домой
• по пути домой

Both are natural. Often по пути домой sounds a bit more idiomatic in everyday speech, but по дороге домой is completely correct and common too.

The main stress is:

По доро́ге домо́й я забегу́ в апте́ку.

Stress matters in Russian, especially for vowels. The important stressed syllables here are:

• доро́ге
• домо́й
• забегу́
• апте́ку

If you want to sound natural, make sure not to stress the wrong syllable in домой or аптеку.