So, this is a really good question that is in many ways extremely difficult to answer for a number of reasons. Some of these are evidence based, we quite simply lack sources from much of the early Middle Ages, and we certainly lack a wide variety of perspectives on the workings of the world from a wide variety of backgrounds and knowledge bases. Indeed, almost all of our sources for the early Middle Ages are from Christian scholars (usually monks) or Christian elites that were written for a Christian audience, so this narrows our sources from the outset. However this does not answer the question really, did these people writing sources, limited as their scope and world views were, have distinct views on these three different phenomena?

The answer is....complex, and it would vary in time and place in the Middle Ages, but lets go ahead and dive on in.

Lets start with magic vs religion. Christian authors of the time, especially monks, are super eager to complain about the prevalence of magicians, soothsayers, and other charlatans conning people into incorrect beliefs if not outright heresy. Most of my expertise is from England at this time, so unless stated otherwise the examples are coming from England, and usually the writings of the Venerable Bede.

Bede for example complains about the number of supposed Christians who wore amulets in an attempt to stave off diseases. He dismisses these attempts at staving off sickness as more or less ignorant superstition, and implies that the amulet wearing quite clearly did not work, but why did he think this? He was certainly not a scientist in the modern sense who observed that the propensity for amulet wearing had no bearing on the number of people who died from disease compared to a control group of those who did not wear amulets. Now his dismissal of this superstitious practice was rooted in his Christian belief which held that magic was quite simply not effective.

Now Bede is a bit of an outlier in his dismissal of the impact of "magic" and there is a good deal of evidence to suggest that the populace at large, even elite members, were quite comfortable with appeals to supernatural powers. Anglo-Saxon England held a good deal of fear of magic wielding witches (as evidenced by their presence in law codes and penitentials of the time) and there is a good deal of surviving material that describes what we would consider magic, ie ritualized incantations, wearing certain items of clothing, and so on. However a medieval Englishman would not have necessarily considered these practices as magical, nor as exclusionary to his Christian faith as Bede did, but as a part of their day to day life. Prayers to certain saints, incantations from the Bible, and other ritualized spoken words were used, as evidenced by their inclusion in medical texts, as a part of the repertoire of Anglo-Saxon medical professionals alongside descriptions of the properties of certain herbs and treatment regimens for conditions such as back pain, blindness, impotence, and so on.

So for certain members of Anglo-Saxon society, magic was an inextricable part of day to day life, for others it was pagan superstition with no actual power.

But what about science? Did science stand astride the petty qualms of the superstitious beliefs of the masses? Well not really.

Science is tricky because it has not been its own discipline of study and knowledge for as long as you might think. Today there is a narrative that science stands in contrast to magic and religion, that was not present, or even really thinkable, in the Middle Ages. Scientific investigation was not its own independent field but something of a supplement most of the time. To go back to our example of Bede, he was a man of God, a monk, writer of Biblical commentaries, histories of the Church in England, and investigations into astronomy and cosmology....?

For example he wrote a good bit about how the Earth was round, the movement of the moon and its effect on the tide, and a whole bunch of other stuff such as the calculation of Easter (an astronomical process even today), the age of the Moon (he was off by a factor of several billion years), the movement of the planets, and so on. His scientific investigation was supplementary, not separate, from his religious investigation, and this pattern is true more or less for the entirety of the Middle Ages. Other evidence of scientific investigation, or at least attempts at it, can be found in medial textbooks that as I described above included commentaries on various medicinal herbs and the correct procedures for certain treatments. However these were included alongside prayers, Biblical passages, and ritualized incantations. The separation of science from religion in the West today is a modern construct not a far reaching or longstanding tradition of opposition, and the two were intertwined in the Middle Ages.

So to sum up, people in the Middle Ages did not rally view science as its own independent discipline. Scientific investigation was certainly still around and medieval Christians were very interested in astronomy and cosmology and how it interacted with scripture and their faith, but it was not viewed in opposition to religious beliefs. Magic and religion is a little bit trickier as different members of society had different perspectives on their relationship. These views ranged from dismissal to condemnation to active usage.

It certainly was the way the world worked. That's a good way to phrase it because quite literally medieval people lived in a world in which curses, prayer, and natural science all coexisted as aspects of the world. As u/Steelcan909 mentioned this is a huge question because our terms don't really apply to the medieval period. People did what we today would call 'magic' and yet they used medical plants and sometimes the sick were healed. So are these people magicians or doctors? And people did what we would call 'science' and yet alchemist distillers occasionally discovered previously unknown elements. So again, are these people magicians or scientists?

The answer is that these concepts were one-and-the-same: Village magic-users were (presumably) devout Catholics. Alchemists used star charts to plan rituals as a part of the distillation process. And sometimes, Catholic priests were secretly black magicians, necromancers even! And sometimes Catholic monks wrote pagan spells and put these pages into a pocket gospel.

The ease at which these three words were conflated is shown in a later medieval document, a guide to household management from Wolfsthurn castle in Tyrol. This book has a lot of useful information on it, in fact let's say "scientific" information... How do you make soap? Prepare leather? Cultivate fields? These questions arise alongside questions about medicine for one's animals but also humans.

Claiming the authority of Aristotle and other learned men from antiquity, the [anonymous] compiler tells how to diagnose fevers, ailments of the eyes, and other medical problems. Further added to this potpourri are prayers, blessings, and conjurations. Medieval people who assembled this and other such manuals would never have thought of themselves as magicians, but the book at hand contains elements of what we can call magic. It recommends taking the leaves of a particular plant as a remedy for 'fever of all sorts'; this in itself would count as science, or as folk medicine, rather than magic. Before using these leaves, one is supposed to write certain Latin words on them to invoke the power of the Holy Trinity, and then one is to say the Lord's Prayer and other prayers over them; this in itself would count as religion. There is no scientific or religious reason, however, for repeating this procedure before sunrise on three consecutive mornings. By adding this requirement the author enhances the power of science and religion with that of magic. - Magic in the Middle Ages, by Richard Kieckhefer (2014), pg. 3

The English intellectual John of Salisbury wrote that in his youth (1130's?) he was being taught Latin by a priest along with a slightly older student. Although the priest deviated from usual instructions, he was using the two students to help him practice divination. It included painting their fingernails and using the reflection for divination. This seems quite difficult, so it also involved a reflective basin. Regardless, in a ritual with the two boys the priest spoke the names of beings which terrified the young John, "By the horror they inspired seemed to me, child though I was, to belong to demons." The older student reported that he saw "certain misty figures" and so these two continued practicing this magic in secret and rejected John (as he had no 'proficiency'). Originally necromancy was summoning the dead to ask them questions (a form of divination), but as early medieval Christians rejected the notion that a magician could actually do this then they realized these conjured beings must be demons. And so that is likely exactly what that priest thought he was doing with his two students, as a later (1400's) handbook on spells/necromancy/demonology from Munich "...contains detailed instructions for conjuring demons in precisely the way that John recounts, and for the same purpose." (Kieckhefer, 151-152)

In a funny twist that Munich demonology book refers to these "magical operations" as experiments, which points to their practitioners naturalistic understanding of their craft (Kieckhefer, 157). And, in between conjurations are "standard" Christian prayers, Psalms from the Old Testament, and invocations of saints (Kieckhefer 161). Why would an aspiring necromancer invoke prayers and saints? Well it is likely because they thought of themselves as a Christian, and perhaps that priest who taught John of Salisbury thought of himself as a powerful Christian as well. So powerful that he had rejected the normal methods of spirituality and surpassed his peers. And so this text could also be thought of as religious, in addition to both magical and scientific.

People in Europe have been walking this line between spirituality and medicine for a long time. Oetzi the Iceman was found in the Alps having died in the late 3000's BCE. His body and kit show that he used multiple "traditions" of medicine. He had a two bundles of birch fungus, as iceman.it mentions this was still used medicinally by people in the 20th century and Rowena Hill has mentioned its possible uses here. And Oetzi had multiple small tattoos of geometric designs at certain points around his body, particularly around joints. Presumably this was done for pain relief and so is a form of magic and the placebo effect. So even thousands of years before the medieval period, people were part magic-user and part natural scientist.

Alchemists knowingly experimented with materials and through the medieval period invented new pigments, such as Sarah Bond mentions: cinnabar was converted to mercury then added with sulphur producing vermillion. You can see an overview of early medieval alchemical texts (and from the other periods) in Laurence Caruana's lecture here. In the words of Lawrence Principe at 12:08 onward:

Alchemy is now recognized as a key part of the history of science. And many alchemists, centuries before the emergence of a modern chemistry, developed sophisticated experimental techniques and based their work and ideas on logical theories of matter and its transformation that they derived from observation and experience. In fact I would argue, alchemy perhaps more than any other scientific practice instilled in early modern Europe a culture of practical experimentation and observation as a means of exploring nature.

As he mentions, alchemy and chemistry were only separated as terms ca. 1700, and William R. Newman here explores the magical transmutation theory pursued by the great 17th century scientist Isaac Newton.

But jumping back to the medieval period, you're asking about the early part of this era. As mentioned, Irish monks in the 800's added pages of spells such as invoking the deity Gobniu to remove a thorn into the back of a 'pocket gospel', see How to get help from a craftsman, by Isolde Carmody. Saint Brigid of Ireland has all the characteristics of the life of an Irish female magician, the various spells she does have parallels as well; but of course she does this magic for Jesus. In the Christianizing early medieval Germany of the 800's some Christians destroyed sacred groves, while other Christians such as the artist of the Stuttgart Psalter at the Wuerttemberg Library depicted Jesus as an armored warrior stabbing/stepping on a serpent and a lion (folio 23) as an illustration of Psalm 91:13. This image must have echoed with western/central Europeans of the period who had a story (as many other peoples around the world do) of a culture hero who conquers a serpent in an epic battle. So religion, magic, and mythology are all being blended in this new Christianized world.

Sometimes these Christians continued practicing as 'pagan' culture bearers, such as the Irish cleric Virgil (at that time working in Bavaria) who was mentioned negatively by Pope Zachary in 748 who says he was teaching that there was another race of men and a world beneath ours, complete with its own sun and moon. The Pope said he should be excommunicated, but presumably Virgil was popular because the opposite happened, he was elected Bishop of Salzburg in 767 and then canonized in 1233 (Celtic Cosmology and the Other World, by Sharon P. Macleod, pg. 15). As you can see, everything was quite blended together in the medieval period.

I've talked about witchcraft and other healing professions briefly here which at the end I've put a bunch of links to information on witchcraft if you're interested in seeing exactly who and what they were doing during this period. They were certainly Christians, as are modern Italian witches as discussed in this amazing paper Italian Cunning Craft, Some Preliminary Observations, by Sabina Magliocco