In the first sentence, at home is an adjective preposition phrase. In the second sentence, home can be used as adverb. In the third sentence, "at home" can be used as adverbial prepositional phrase. Am I right? Are the three sentences correct? Thank you.
All three sentences are correct English. Exactly how you classify the parts of speech and grammatical structure is debatable (seriously, linguists debate this stuff).
However, sentence 2 and 3 have very distinct meanings, and actually make use of different senses of the verb to take.
I usually take my breakfast home
Usually is an adverb, take is the verb, my breakfast is the direct object, and home the indirect object. The speaker is talking about moving their breakfast to their home.
I usually take my breakfast at home
This uses an alternate sense of the verb take that is applied largely to food and drinks, though also exists somewhat similarly for things like tests and exams. At home is a prepositional adverb phrase indicating where the speaker takes his breakfast. To take tea is to have a drink of tea, possibly with accompanying light foodstuffs, and one might take wine with one's meal - as well as find gentlemen of prior centuries telling their family that they will be taking their dinner at their club. In the sense of food and drink, it could usually be replaced with have in the modern vernacular usage, or eat or drink as appropriate. In this case, the speaker is saying that they have breakfast at home, rather than away from home. The my is essentially extraneous, and does not affect the meaning.
