For the early Middle Ages, say into the 11th century, it's really hard to get an idea of how much the average lay person knew, how frequently they attended church, what the practice of confession looked like and how frequently it occurred (this will be important later), what the education level of the priests serving them even was. There's an infamous case of a letter exchange where bishops conducting a visitation worry that the local priests are committing heresy by baptizing in the name of the Father and the Daughter and the Holy Spirit, because they're mixing up their Latin declensions.
The 11th century and then the 12th in particular see an explosion in religiosity among the nobility and then the upper peasantry or proto-urban gentry. But this is most visible in people whose families already have money and status reforming existing religious communities, joining more dedicated ones--or founding new orders.
Indeed, the 12th century is the first time we can really talk about different monastic orders in the West. Especially important for our purposes here are the Augustinian canons, who were founded on the idea of a more pastoral or outreach approach. But this is still an order tied to existing churches in towns; we're still not at the level of rural parish priests preaching and teaching yet. In the 12th century, we do see the rise of some famous traveling preachers, or religious celebrities who embarked on preaching tours like Bernard of Clairvaux and Hildegard of Bingen. Both of them are known to have preached to laity, but what level of laity? Probably still not the average serf.
The rise of Gothic art and architecture would have brought a new mode of religious instruction to people lucky enough to live near new construction and have decent distance vision. That's not to say earlier churches were bare and devoid of art! But later medieval architecture built in religious instruction (stories, pictures of hell and purgatory) to its impressive stained glass windows and stone carvings all over the place.
The later 12th century and early 13th is a wake-up call for the Church. The 1215 Lateran IV (council) marks the first really big turning point in terms of Church attention to the laity. The council decrees that all Christians of both sexes must take the Eucharist once and year, and confess once a year. Confession is important: it offered a chance for a priest to meet one-on-one with each layperson and, as best he could while mindful of all the other people waiting in line for their turn and jockeying to get the gossip on what everyone else was confessing (there is no private "confessional" until after the Reformation; here you knelt before the priests with everyone else looking on andlistening), correct misconceptions the penitent might have about religion.
Mostly, we know this consisted of moral behavior, the virtues and vices. These were certainly considered religious, but there would also be questions about skipping Mass, making semi-sacrilegious jokes, venerating saints improperly, and so forth. In this way, confession did offer a chance for religious instruction.
You're right that Mass was said and hymns were sung in Latin, although it's more than probable that over the course of your lifetime you'd gather the gist of things. More importantly, however, was preaching. Although surviving sample sermons are mainly in Latin, there are hints they were preached in the vernacular starting pretty early on. And a big concurrent or post-Lateran IV development was the rise of the preaching orders: the Franciscans and Dominicans. These friars had the explicit mission of preaching to the laity. Medieval sermons did still focus on moral topics, but they used Bible verses and stories as well as legends of saints and other religious exempla (anecdotes) to ground the lessons.
The late 14th and then the whole 15th century witness the next great wave of clerical attention to the laity. This is the era of even more preaching and more public demand for more and better sermons, councils mandating more frequent confession, increased religious zeal among basically all levels of the population visible in the sources (which, still, it can be harder to "see" rural peasants). Although the overall literacy rate would still have been low by 1500, maybe 10-15% at most, the catechetical literature that was enormously popular among lay people who could read gives us an idea of what people would have known: the Trinity, the Our Father and Hail Mary, the Creed, the 10 Commandments, vices and virtues, the meaning of the sacraments especially the Eucharist. Saints' lives were popular literature, dramas, and anecdotes in sermons, too. People went on pilgrimages to local holy sites and saints' shrines, so they would have learned those stories and thus the saintly mediation-Jesus' salvation in that way, too.
But for the benefit of literate and illiterate laity alike, the 15th century sees MUCH more attention to educating rural parish priests so they can teach their parishioners. Handbooks of pastoral care are much more frequent in this age. Preachers have quick reference guides on how to interpret and explain difficult Bible verses in vernacular French and German! So by the end of the Middle Ages, sermons and confession came to serve as a way even illiterate rural peasants could know their basic catechism and achieve their salvation.
That's from the clerical perspective. What can we see about religious enthusiasm from the laity, beyond the nobility patronizing new orders in the 12th century? This is one area where France is a really interesting case study. First, the southern France/northern Italy kind of crescent or blob witnesses several major lay or quasi-religious (meaning, like a religious order but not officially recognized by the Church) movements that the Church identifies as heretical. It's difficult or impossible to know what the heretics actually believed, since our major sources are inquisitors and inquisitorial paranoia (basically, a tendency to view all heresy in light of ancient Arianism and Manicheanism).
But it seems evident from confessions under torture that these 'heretics'--who admittedly were among the more religiously astute/educated/eager, even if their education wasn't always right--generally had the basics of Christian doctrine like the Trinity understood, even if they had a different interpretation of some of it. (Like confessing a belief that a particular local figure was the embodiment of the Holy Spirit, for example). So in cases like that, we can see that individual people and groups certainly sought out knowledge and possessed a fair bit.
Second, the Great Schism at the end of the 14th/early 15th century has a MASSIVELY different impact on the European population than earlier schisms. Renate Blumenfeld-Kosinski has an excellent study where she shows how the earlier mainly-German messes really only affected, in terms of who writes about them, monastics and bishops directly in the schismatic territories in dioceses and monasteries closely linked to the highest power lines. The Western Schism, on the other hand, with the papacy split between Avignon and Rome and all of western Europe choosing sides, has a much deeper impact. Lay people--mostly urban gentry, but still well beyond our aristocratic nuns and monks--issue prophecies and visions about which side will triumph, which side should triumph, the devastation that is being wrought on Christendom. Perhaps the average serf in the north of France is much more affected by Burgundians and English soldiers torching their fields in the ongoing Hundred Years' War, but the religious conflict--even at the level of papal politics!--is significant enough both religiously and publicly that lay people carve out a way to have their say.
Third, we can finally see individual, rather spectacular cases like Joan of Arc. She was a peasant, although a well-off one (her parents were landowners), and certainly she was more religiously inclined than average. Nevertheless, she wasn't particularly well-educated in religious or other matters. And the answers she gave at her trial still demonstrate a nice awareness of both religious dogma and the religious devotional culture of her time, like awareness of saints' lives and the importance of the devotion to the Name of Jesus. She knows, too, how to maintain and defend her religious orthodoxy-- which is a significant achievement, because it means she knew what would decisively condemn her as a heretic. (I mean, obviously she was condemned, but probably nothing she herself said would have mattered.)
So over the course of the Middle Ages, the Church paid an increasing amount of attention to the spiritual needs and education of its laity. The impact on the rural peasantry was the slowest to grow, but even it was taking hold by the end of the era. It's possible to see the Church actually responded, bit by bit, to increased religious enthusiasm on the part of lay people, whether Lateran IV is a response to perceived 12th century heresy or the 15th century movement to pastoral theology responds to the increasing lay attention to religion and religious politics in the wake of Schism and warfare. In general, though with obvious exceptions, people were increasingly interested and increasingly knowledgeable; the Church sought to be increasingly attentive and increasingly able to provide that knowledge.
The Middle Ages was a deeply and inherently religious era, even if priests and people didn't always know their "of the Son" from their "of the Daughter."