The problem with this question is the fact that "the Middle Ages" spans a vast period of time: it is generally used to refer to the whole period between 500 and 1500 AD. So what may be true in, say, 750 AD is unlikely to still be the case in 1485 AD.

As others have noted, magic was not a belief system but was rather something that virtually everyone believed existed and which could be used for good or for evil. As Richard Kieckhefer notes in his excellent study Magic in the Middle Ages, the line between a prayer and a magical invocation or between a charm and a blessing was blurred at best and almost all forms of Medieval magic were not only entirely accepted by Medieval clergymen, but were also widely practiced by them. Magic based on the invocation of demons was definitely considered a very bad thing and in the later Middle Ages and early Modern Period there was a shift in theology which cast all magic under suspicion, but for most of the Medieval period it was not magic that could get you in trouble but what you did with it.

"Pagan beliefs" are another matter. Firstly, there is a romantic idea that Christianity was simply a veneer imposed on the mass of people who either simply paid lip service to it or rejected it outright and were "really pagans" underneath. As common as this idea is, encouraged by the modern neo-pagan movement and often by Hollywood, it's largely nonsense. Pagan practices lingered on for centuries, but mainly only as folklore and traditions, not as underground survivals of any belief system. So a villager in Abbots Bromley in Staffordshire might take the ancient antlers from the church and perform the "Horn Dance" on the first Sunday in September, as his pagan ancestors had for centuries, but he would have virtually no idea what on earth this dance had meant to those pagan ancestors - it had just become a traditional ritual.

As other answers have noted, while whole pagan belief systems did not survive, many pagan beliefs became absorbed into Medieval Christianity. The idea of venerating saints, praying to them and keeping their relics as holy objects began as a purely Christian practice - to honour devout people, especially those martyred during the Roman persecutions. But once established, this cult of the saints took on most of the trappings of the earlier pagan polytheism, with the saints taking on the roles of various minor and local gods and spirits in paganism and often with the local saint being venerated in the same place as a former local god or goddess. Often this process went full circle, with local gods being turned into "saints": Saint Christopher and Saint Brigid both began as pagan deities.

On the whole though, despite this and despite surviving pagan folklore, people did consider themselves Christians and could be, in their own often unsophisticated way, highly devout. Religion in the pre-Modern world was a communal business, not a matter of private conscience. People lived and shared their faith as communities. This is why the history of the conversion of Europe to Christianity is one of the conversion of communities and groups and rarely one of the conversion of individuals. The individuals we do hear of converting are kings and warlords and we then hear of their subjects and followers doing the same.

This is often taken by modern people as evidence that these mass conversions "were not genuine", since modern religious beliefs are personal and private and unlikely to be swayed by the fact that a leader has converted. That represents a total misunderstanding of pre-Modern religious ideas. In the pre-Modern era, the conversion of a powerful war leader or king was seen as an indication that the old gods no longer had sufficient power and that this new faith was better protection for the community. Since we don't live in a world of subsistence-level agriculture, potential famines and the prospect of raiding armies at any time, the idea of religion as protection in a hostile environment is alien to us and we see it purely as a source of personal comfort and satisfaction. Pre-Modern people didn't have that luxury.

So what evidence we do have of popular religion in the Medieval period shows that the bulk of people were generally devout in their own way. This religious devotion could take on a fierce and sometimes violent intensity. In the Tenth Century it was the peasantry that propelled the "Peace of God" movement that began to rein in the predatory warfare of many local lords. And the Crusading movement quickly became a mass movement of considerable power and intensity, rather to the surprise of the very clergy who preached the idea in the first place. As wrongheaded and often misdirected as peasant Crusading enthusiasms like the Peasants' Crusade and the so-called "Children's Crusade" were, there is no doubt they were driven by genuine faith. And on the First Crusade the squabbling of its noble leaders was repeatedly ended by "the people" telling the aristocrats to remember their holy mission and, on one occasion, informing them that they would be abandoned so the bulk of the army could continue to Jerusalem without them.

So Medieval peasants didn't sit around debating Anselm's ontological proof of the existence of God or the ins and outs of the doctrine of Transubstantiation. And their level of active piety could be as patchy as their grasp of doctrine. But the idea that they were all secret pagans or that their Christianity was simply paganism in disguise is largely nonsense.

See Richard Kieckhefer Magic in the Middle Ages
Richard Fletcher, The Conversion of Europe: From Paganism to Christianity 371-1386 AD
Rosalinde B. Brooke & Christopher Brooke, Popular Religion in the Middle Ages: 100-1300
Bernard Hamilton, Religion in the Medieval West