Learn Hardcore Russian: Зайди в аптеку за лекарством по дороге домой, пожалуйста. - Please stop by the pharmacy for some medicine on your way home.

Зайди is the singular informal imperative form of the verb зайти (perfective), meaning to drop in / to stop by (briefly).
The base verb is идти (to go), and за- adds the idea of a short stop or going “in” as part of a route.

What’s the difference between зайди and заходи?

зайди (perfective imperative) = do it once / make a single stop: Stop by the pharmacy (this time).
заходи (imperfective imperative) can mean: 1) come in / drop by (more generally, habitually, or whenever): Drop by sometime.
2) sometimes a “come in” invitation while someone is at the door: Come in.

In this sentence, the “one specific errand” feeling fits зайди.

Why is it в аптеку and not в аптеке?

Because в + Accusative is used for motion/direction into a place: в аптеку = into/to the pharmacy.
в аптеке would be location (“in the pharmacy”), using в + Prepositional.

Why does аптека become аптеку?

аптека is feminine singular. After в with motion, the noun takes the accusative case:

Nominative: аптека
Accusative: аптеку

Why is it за лекарством and not за лекарство? What does за mean here?

Here за + Instrumental means to go to get something / to pick something up.
So за лекарством literally means for medicine (to fetch it).

Important: за has multiple meanings. Another common one is behind (e.g., за домом = behind the house), but with verbs of motion it often means go and get.

What case is лекарством, and why that case?

лекарством is instrumental singular of лекарство.
After за in the “go get/pick up” meaning, Russian uses the instrumental:

за + Instrumental = go for / go to fetch

So: за лекарством = to get medicine.

What does по дороге домой mean grammatically? Why по?

по дороге домой means on the way home.

по + Dative often means along / by / on (a route, process, schedule).
дороге is dative singular of дорога (same form as prepositional here, but it’s functioning as dative with по).
домой is an adverb meaning (to) home (direction).

So: по дороге домой = while traveling home / along the route home.

Why is it домой and not дома or к дому?

домой = to home / homeward (direction; very common with “go/come” contexts)
дома = at home (location)
к дому = toward the house (often more literal: up to the building; not as idiomatic as домой for “go home”)

In “on the way home,” Russian naturally uses домой.

How polite is this sentence, and how would I say it formally?

As written, Зайди …, пожалуйста is informal (to one person you address as ты).
For a formal or plural version (to вы or multiple people), use:

Зайдите в аптеку за лекарством по дороге домой, пожалуйста.