In terms of direct evidence for Augustus himself, we only have a very short description of his education in Suetonius (Life Of Augustus 84):

"From early youth he devoted himself eagerly and with utmost diligence to oratory and liberal studies. During the war at Mutina, amid such a press of affairs, he is said to have read, written and declaimed every day."

Suetonius also describes Augustus' literary tastes, from which we can infer some further information about his education (Life of Augustus 85):

"He wrote numerous works of various kinds in prose, some of which he read to a group of his intimate friends, as others did in a lecture-room; for example, his "Reply to Brutus on Cato." At the reading of these volumes he had all but come to the end, when he grew tired and handed them to Tiberius to finish, for he was well on in years. He also wrote "Exhortations to Philosophy" and some volumes of an Autobiography, giving an account of his life in thirteen books up to the time of the Cantabrian war, but no farther. His essays in poetry were but slight. One book has come down to us written in hexameter verse, of which the subject and the title is "Sicily." There is another, equally brief, of "Epigrams," which he composed for the most part at the time of the bath. Though he began a tragedy with much enthusiasm, he destroyed it because his style did not satisfy him, and when some of his friends asked him what in the world had become of Ajax, he answered that "his Ajax had fallen on his sponge"."

From these two sections it's fairly clear that Augustus was educated in oratory, philosophy and literature, certainly in both Greek and Latin. If he was writing a trajedy about Ajax he must've read Sophocles and Homer, and if he was writing about philosophy then he probably also read the Greek masters: Plato, Aristotle etc. In this he would have been fairly typical of young men of his social class, all of whom would have received schooling in these subjects. Saying anything more about Augustus himself is speculation, but we can say something about elite education in late Republican Rome and tentatively apply it to him.

Cicero gives a fairly detailed account of his oratorical training in the Brutus (301ff.), and also discusses education in this section of de Oratore (1.158-9):

"The poets must also be studied; an acquaintance must be formed with history; the writers and teachers in all the liberal arts and sciences must be read... The civil law must be thoroughly studied; laws in general must be understood; all antiquity must be known; the usages of the senate, the nature of our government, the rights of our allies, our treaties and convention, and whatever concerns the interests of the state, must be learned."

Cicero says that he learned from the best men in the city at the time - he trained with the top orators and legal minds, and studied with a number of Greek philosophers living in Rome. He also travelled to the Greek world to further his studies, including Athens. Whether the same can be said for Augustus is uncertain, but there's no doubt that his family connections would've made it possible.