Among the best primary sources for this are Institutio Oratoria by Quintilian (c.35 to c.100 BC) and Dialogus de oratoribus, usually attributed to Tacitus (c.56 to c.100). Both of these are cited extensively by Stanley Bonner in Education in Ancient Rome: From the Elder Cato to the Younger Pliny. It should be noted, though, that the liberal education referred to by Tacitus in his Agricola was by this time seen by both Quintilian and Tacitus as being less widely taught than in the time of Cicero.
Subjects
In addition to Latin, Greek, grammaticus (including literature / poetry), mental and moral philosophy and rhetoric and oratory, a liberal education would have included (though perhaps not all of them) history, law, arithmetic, geometry, musical theory and astronomy (the last four usually came together). Other subjects may well have been included, depending on the circumstances of the pupil / student and the preferences of the parents. Quintilian, in Book 1, gives grammaticus a simple, initial definition but then goes on to cover much more, showing how intertwined many of the subjects are:
This profession may be most briefly considered under two heads, the
art of speaking correctly and the interpretation of the poets; but
there is more beneath the surface than meets the eye. (3) For the art
of writing is combined with that of speaking, and correct reading
precedes interpretation, while in each of these cases criticism has
its work to perform....(4) Nor is it sufficient to have read the poets only; every kind of
writer must be carefully studied, not merely for the subject matter,
but for the vocabulary; for words often acquire authority from their
use by a particular author. Nor can such training be regarded as
complete if it stop short of music, for the teacher of literature has
to speak of metre and rhythm: nor again if he be ignorant of
astronomy, can he understand the poets; for they, to mention no
further points, frequently give their indications of time by reference
to the rising and setting of the stars. Ignorance of philosophy is an
equal drawback, since there are numerous passages in almost every poem
based on the most intricate questions of natural philosophy....(5) No small powers of eloquence also are required to enable
the teacher to speak appropriately and fluently on the various points
which have just been mentioned. For this reason those who criticise
the art of teaching literature as trivial and lacking in substance put
themselves out of court.
By the mid to late 1st century AD, there was a heavy emphasis on declamation in rhetoric. Bonner explains declamation:
Originally, this term referred simply to the voice training process,
but before the end of the Republic, it was extended to mean the
rhetorical practice-speech itself. Then, in Latin, any kind of
rhetorical speech on a stock theme came to be termed a 'declamation',
whether it was delivered in loud tones or not, and the speakers were
called 'declaimers'.
Tacitus was particularly scathing on the decline of eloquence and substance in speech (he blamed none other than Seneca for starting this trend), at the same time describing what he saw as a good 'liberal education':
No man was ever yet a complete orator, and, I affirm, never can be,
unless, like the soldier marching to the field of battle, he enters
the forum armed at all points with the sciences and the liberal arts.
Is that the case in these our modern times? The style which we hear
every day, abounds with colloquial barbarisms, and vulgar phraseology:
no knowledge of the laws is heard; our municipal policy is wholly
neglected, and even the decrees of the senate are treated with
contempt and derision. Moral philosophy is discarded, and the maxims
of ancient wisdom are unworthy of their notice. In this manner,
eloquence is dethroned; she is banished from her rightful dominions,
and obliged to dwell in the cold regions of antithesis, forced
conceit, and pointed sentences. The consequence is, that she, who was
once the sovereign mistress of the sciences, and led them as handmaids
in her train, is now deprived of her attendants, reduced,
impoverished, and, stripped of her usual honours (I might say of her
genius), compelled to exercise a mere plebeian art.
Texts
Quintilian, who published Institutio Oratoria in 95 AD after 25 years as a teacher, lists numerous authors in Book 10 used in teaching: Cicero, Demosthenes, Plato, Aristotle, Thucydides, Livy, Theophrastus, Homer, Hesiod and Virgil were obviously among them, but he mentions many more, and also writes of those he doesn't mention by name:
I am well aware of the existence of the poets whom I pass over in
silence, and am far from condemning them, since I have already said
that some profit may be derived from every author.
