By the late Middle Ages and Renaissance (so, roughly 1400-1600), Europe was bursting with churches and priests. There’s a simple reason: Europe was overwhelmingly Christian, and every Christian (besides professed religious attached to a religious order and living communally and a few other exceptions) was bound to a parish by canon law (the law of the church). Your parish was where you received the sacraments that identified you as a Christian. In other words, for the vast majority of people, it’s where they were baptized, confirmed (if they were confirmed), went to confession to their parish priest or his assistant priests, fulfilled their obligation to receive Holy Communion at least annually, went to Mass weekly or more often, had their marriage blessed by their priest, received the Last Anointing on their death bed, and were buried. It’s also where they were legally obliged to pay tithes—a tenth of anything their income—to the rector of the parish (who, by this period, might be a single individual or, more likely, some religious corporation, like a monastery or college). Baptismal rights, burial rights, and entitlement to tithes were part of the legal definition of a parish. The individual or corporation that held these rights bore the “cura animarum” or “care of souls” for parishioners, which was often by this time exercised by a deputy priest (called a vicar in England) who was paid a salary from the income from tithes and land.

By this time, there were thousands of parishes scattered across cities, towns, and villages. Already by the late 1200s, London, the largest city in England, had over 100 parishes. The city of Norwich in England with a population of c. 10,000, one of the largest towns in the country, had about 40 parish churches. Later medieval England had around 9,000 parishes. On the continent, Paris, the largest city in Europe seems to have only 35-40 parish churches for its population of around 200,000 in 1328. Smaller towns like Bourges and Poitiers had 20 or 30 parishes by the 11th century. There tended to be fewer parish churches per town in German regions, in part because towns there often grew around older, established parish churches. As George Martin implied, there were, of course, more rural areas where parishes might be few and far between, I’m thinking of parts of Wales or Spain. There are early 14th-century complaints from Welsh parishes that their parish priest does not regularly fulfill his duty to visit them every Sunday. But in general, the majority of Christians were in easy reach of a church, so much so that loyalties to one’s parish could be intense, especially in Italy where we speak of “campanilismo,” absolute devotion to and pride in your neighborhood, defined by what was in sight of the neighborhood parish church’s bell tower.

As you noted, there were also chapels, small annexes to a church, side altars in a church, or a small freestanding church at some distance from a parish church but dependent on it. A chapel of-ease might be built if your rural parish was large, and getting to the main church posed problems during cold or wet weather. In Scandinavia, there were hostel-like establishments—I’m blanking on their name—attached to churches where people coming in from a distance could stay overnight before traveling back to their homes after attending their parish church. Wealthy people could obtain license to have a private chapel on their estates. But parish churches carefully guarded their rights to tithes, administering baptism and confession, and burial rights against any encroachments from chapels-of-ease or private chapels. By the later Middle Ages, there would also often be a chantry chapel or chapels inside the parish church where families hired priests, called "chantry chaplains," to celebrate (i.e., sing or chant-thus the name) private masses for the souls in purgatory of those families who hired the priest.

As for priests, since every parish needed at least one priest and there were thousands of parishes, there were even more clergy. By this period, it seems you could barely turn around without running into one. They were such familiar figures that the great literature of the period—Chaucer’s Canterbury Tales, Langland’s Piers Ploughman, Boccaccio’s Decameron, Dante’s Comedy—could satirize their well-known foibles. Every parish had a rector who, if not resident himself, had to find a deputy to attend to the spiritual rites and needs of his parishioners. I mentioned private chantry priests who were also growing in numbers at this time; during their free time, they might teach some elementary education to village children. A larger parish would have assistant priests, deacons, and subdeacons to aid the rector or his vicar. By this time, there were also thousands of priests belonging to the various mendicant orders that arose in the early 13th century—Franciscans, Dominicans, Augustinians, etc.—many of them charged to wander from place to place preaching (their specialty) and hearing confessions; after 1300 bishops were required to license a certain number of mendicants to administer sacraments in their diocese. Some orders, like Augustinians, actually ran some parishes. In some cases, a local monastery might be the area’s parish church. Just as lay society broke into hierarchies, so did the clergy. Wealthy cardinals, bishops--often servants to royal government—and well-beneficed priests (i.e., those who held title to more than one parish) were the apex. Then came priests who actually served the daily needs of parishioners when their rector was non-resident; they were often badly paid as were the parishes assistant clergy. This “clerical proletariat,” as the great church historian Pantin called them, were the unbeneficed priests who did the grunt work of the parish. Like Europe’s parishes, they numbered in the thousands.

Sources: Though wider in time span and a little more specialized than you might want, C. N. L. Brooke’s Churches and Churchmen in Medieval Europe (1999) is a good starting point. You can browse a free preview at Google Books. Daniel Borstein edited a useful survey of several topics above in Medieval Christianity in A People’s History of Christianity (2010).