По дороге домой я решил заехать в аптеку за лекарством.

This is a very common Russian expression.

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

Together, по дороге домой means on the way home.

Russian often uses this kind of compact phrase where English would prefer a fuller expression.

Дороге is in the dative singular.

The preposition по often takes the dative when it means things like:

• along
• according to
• during / in the course of
• in set phrases like on the way

So:

• дорога = road, way
• по дороге = along the road / on the way

This is something learners usually just get used to as a fixed pattern: по дороге + destination.

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

Compare:

• Я иду домой. = I am going home.
• Я дома. = I am at home.

In this sentence, the person is not already at home; they are traveling in that direction, so домой is the correct choice.

Because решил means decided, and in Russian, verbs like decide, want, plan, can, must, etc. are commonly followed by an infinitive.

So:

• решил заехать = decided to stop by

This works much like English decided to + verb.

Here заехать means to stop by somewhere on the way, usually briefly and often while traveling by some means of transport.

It comes from the verb ехать = to go by vehicle / ride / drive.

So заехать often has the sense of:

• drive in
• stop by
• drop in
• swing by

In this sentence, it suggests the speaker planned to make a quick stop at the pharmacy while on the way home.

Заехать is a perfective verb, and that fits well here because the speaker decided to do one complete action: make a stop.

Russian often uses perfective verbs when talking about:

• a single completed action
• a planned result
• a specific event

So решил заехать sounds natural because it refers to one concrete stop at the pharmacy, not a repeated or ongoing activity.

Its imperfective partner would usually be заезжать.

Because в аптеку shows motion toward a destination.

Russian uses:

• в + accusative for motion into/to
• в + prepositional for location in/inside

So:

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

Since the speaker decided to go there, the accusative is required.

In this sentence, за means to get / to fetch / for the purpose of getting.

This is a very common Russian pattern:

• идти за хлебом = to go for bread
• зайти за другом = to stop by for a friend / pick up a friend
• поехать за покупками = to go shopping

So заехать в аптеку за лекарством means to stop by the pharmacy to get medicine.

This use of за is different from meanings like behind or beyond.

Лекарством is in the instrumental singular.

After за in the sense of to get/fetch something, Russian uses the instrumental case.

So:

• лекарство = medicine
• за лекарством = for medicine / to get medicine

This is simply the normal grammar pattern:

• за + instrumental

Yes, it could be plural in another context, but singular is natural here.

Лекарство can function a lot like medicine in English: it often behaves as a general or mass-like noun when you mean medication in a general sense.

So:

• за лекарством = to get some medicine / medication

If you meant several specific medicines, you might say:

• за лекарствами = for medicines / medications

Both are possible, depending on context.

Russian word order is flexible, and the beginning of the sentence often shows what the speaker wants to emphasize or set as the background.

Starting with По дороге домой puts the setting first:

• On the way home, I decided...

That sounds very natural because it gives the situation before the main action.

If you said:

• Я решил по дороге домой заехать в аптеку...

that would also be correct, but the focus would feel a little different.

Yes, in some contexts it could be omitted, because the verb решил already shows first person singular masculine.

So depending on context, Russian could say:

• По дороге домой решил заехать в аптеку за лекарством.

This is common in conversational Russian, especially when the subject is already obvious.

However, keeping я makes the sentence clearer and more neutral, especially for a standalone sentence.

They express two different things:

• в аптеку = the destination
• за лекарством = the purpose

So the structure is:

• stop by where? → в аптеку
• for what? → за лекарством

This combination is extremely normal in Russian. English often does the same:

• stop by the pharmacy for medicine

So it is not redundant; it gives both place and purpose.