In particular one chapter of A Clash of Kings from Tyrion's POV detailed the order of battle between a Stark and a Lannister army. Tyrion was placed in the weaker Lannister ranks which he notes were mostly composed of peasant conscripts who were barely armed, often having not much more than farming tools with little to no armor and no battle training.
Did this happen much in Early Medieval armies? I know in the Early Middle Ages that there wasn't much of a standing army and that for large campaigns nobles usually levied or conscripted peasants, but to what degree were they armed and trained?
Large numbers of conscripted peasants were not common in Medieval warfare. Where they were sometimes used was in defense. They were rarely used in offense.
The make up of Medieval Armies, including those of France and England varied by time period through the Middle Ages (and by geography).
From say 1100 to 1300, the backbone of a French or English medieval army was the "feudal host". This was made up of the knights which the tenants in chief owed the crown in return for their land holdings.
It is important to remember that "knight" does not just mean one man. A "knights service" usually included an armed, armored and horsed knight, one or several armed and horsed squires, several pack horses, and up to half a dozen grooms and other servants (all generally semi-trained and equipped as infantry soldiers).
There could also be "household knights" who were landless knights, supported (paid) by the king or other lords (rather than serving as a duty of their land tenure).
Infantry, for the offence, did not generally include untrained peasant levies. Not worth the expense of sending such dubious troops on campaign. Infantry (beyond that which came with the knights) would be recruited (from fit people ideally with some military experience).
On defense, the army was more likely to call up peasant levies, especially if defending fortified positions like towns or castles. These would generally be called up from the immediate area under threat, rather than some sort of national or widespread peasant levee.
After (about) 1300 (perhaps earlier in England than in France), personal military service by knights was replaced more and more by a money payment in lieu of personal service. So the backbone of the army became retained knights (or, increasingly, "men at arms" or "sergeants" who were horsed, armored and armed like knights, but who had not been knighted) who were paid.
It was also in about 1300 that the English began to rely on longbowmen as an important component of their armies. Although all Englishmen were first encouraged, and then instructed to train with the longbow, the longbowmen in the armies were recruited and paid, not drafted as part of a peasant levee. (Of course, if England had been invaded, a levee en masse of longbowmen might have been likely and formidable).
As payment for military service became more common, more and more of an army was made up of "professional soldiers" and some of these formed mercenary companies to collectively hire their services to armies.