Learn Russian: По пути домой он купил хлеб.

Путь is a special "anomalous" noun. Even though it is masculine, in the genitive, dative, and prepositional cases it declines exactly like a feminine noun ending in -ь (such as тетрадь). Because the preposition по takes the dative case, путь becomes пути.

Is по пути just a set phrase?

It translates neatly as "on the way", but grammatically it is just the preposition по (meaning "along") paired with the dative case of путь (way/path). It's an extremely common combination used whenever someone does a secondary action while traveling somewhere.

Why do we use домой instead of дома?

Домой is an adverb of motion specifically meaning "homeward" or "to home." Since "on the way" implies movement towards a destination, you must use домой. You use дома only when describing a static location ("at home").

Why use the perfective verb купил instead of the imperfective покупал?

The perfective купил is used because this is a single, completed action that yielded a result—he successfully got the bread. If you used the imperfective покупал, it would mean he was in the middle of the buying process, or it could mean he habitually bought bread on the way home.