I'm going to take the question to mean Western and Central Europe, specifically. If you want a broader answer or for other parts of the world, I unfortunately won't be much use.
While the specifics would vary widely based on your level of education and the culture in particular, there are some safe general assumptions that we can make. For starters, it would be helpful to envision the universe as one gigantic, organic whole, pulsing with natural, preternatural (not-quite-supernatural, but still wondrous and unknown all the same), and spiritual energies that have domains on Earth and in the heavens; while we could easily perceive the effects of these energies, we really had no idea as to their ultimate cause. This, in fact, is the very definition of "occult," and the world was full of it.
Since the instrumentation didn't exist to understand things like, say, magnetism or disease, physical phenomena were understood largely relationally, mostly in terms of sympathies -- that is, things that happen on Earth have direct cosmic linkages (and vice-versa). This is best explained by the famous line of the Emerald Tablet, "that which is below is like that which is above, and that which is above is like that which is below..." and served not only as the foundational basis for why astrology, alchemy, and magic "worked," but also provided a comprehensive understanding of the universe itself.
In many ways, astrology undergirded much of this. The patterns set in the sky naturally gave way to mathmatic ratios that were generally understood in terms of music ("the music of the spheres"). This music, while not something you or I could listen to, had profound influences on what happened on Earth. Plagues, famine, good harvests, political upheaval, and just about anything else could ultimately be traced back to the abstract spiritual effect that this music had on the world. The alignment of the stars and planets at the time of our births also established our baseline humoral disposition, which affected everything from our personalities to our physical health (we were, after all, microcosms of the universe, so it stood to reason that macrocosmic shifts in the stars would have some effect on us).
In alchemy, which was understood as a sort of terrestrial astrology, many natural philosophers by the 17th Century understood matter to be based on a very specific ratio of abstract particles -- Mercury, Sulfur, and Salt. These weren't the specific elements that we know today, but were rather placeholders for specific physical properties possessed by material objects. Since the physical world provided countless examples of how these ratios can be altered -- burn a piece of paper and you'll immediately see this -- it stood to reason that humans could artificially and permanently transform matter from one form to another, with the most perfect form of matter being gold itself. This had clear linkages to astrology, as every planet was keyed to a specific metal and served as a kind of physical microcosm of celestial changes.
As for magic, occultists of the 16th and 17th centuries largely concerned themselves with astrological magic (that is, understanding the mathematic ratios of celestial bodies and tying that to a larger philosophical framework, generally Kabbalah or other schools of Hermetic thought) or using ritual to tap into hidden natural and spiritual forces to bring about a desired change in the physical realm. This truly runs the gamut, and I won't go into specific schools of magical thought here, but suffice it to say that the sixteenth and seventeenth centuries saw a concerted effort by some of the brightest minds in Europe to document occult forces and express those forces mathematically (as that was, for all intents and purposes, the basic language of magic).
It also informed medicine and what we'd call pharmacology, as any ailment could be keyed to astrological phenomena, which in turn had a profound influence upon the body's humoral system. Diseases were, therefore, a question of humoral imbalance rather than external attack (as we understand them now). While there was vigorous debate in the late 16th through 18th centuries about the nature of treatments, what any medical practitioner would understand is that a patient's humoral imbalance was likely caused by certain astrological alignments which messed around with the body's natural humoral balance. Restoring balance, therefore, required the consumption of foods, plants, and metals that had specific affinities for the astrological signs that pointed towards rebalancing the patient (either through directly canceling the effect out with its elemental opposite or by overloading the elemental imbalance to the point of provoking the body to purge itself).
I want to stop here and throw out some major caveats to what is a very complicated question.
First off, scholars were by no means an agreement on much of this. There were extraordinarily diverse schools of thought amongst occultists, medical practitioners, and alchemists, and the seismic shifts in astronomy certainly didn't go ignored by astrologers. The intellectual millieu of the 17th century, in particular, was one that was virtually obsessed with hidden meanings, the power of symbols and emblems, and the necessity of keeping knowledge secret. Practically speaking, this meant that knowledge -- and the means of acquiring it -- could be radically individualistic and were largely tied to the individual's own moral standing.
Second, notice how I've only really talked about fields that would have been exclusively the purview of the well educated. The peasantry and the working class would have digested the general world view of sympathies and the unity of the universe, but their direct experience with it would have been within the context of folk magic and local healers.
Third, it's important to note that what we would consider "science" now was one of many different schools of thought during this period. Why experimentalists, coupled with a philosophical understanding of the universe that was mechanistic (read: predictable and knowable) and could be expressed symbolically through mathematics is well beyond the scope of your question, although I highly recommended reading Leviathan and the Air-Pump, by Steven Shapin and Simon Schaffer, for a good starting point to answering this question.
Fourth, the very nature of this cosmological worldview was that the pursuit of knowledge was necessarily interdisciplinary and required the scholar to be well-traveled. Alchemists often found jobs as mining engineers, gem cutters, metallurgists, economists (for a lack of a better term), and practical chemists (like Johann Becher); any doctor worth their salt would be well-versed in astrology and emblem magic; and occultists were often some of the leading lights in applied mathematics, geography, and pedagogy (John Dee immediately comes to mind).
Finally, I've done my best to use the term "natural philosophy" in lieu of "science" when discussing this era. It's important to realize that the pursuit of knowledge was not for isolated facts that could be repeated, like it is now, but rather to bring about physical effects which could explain larger philosophical truths derived from reason alone. Again, the historiography is rich on why the change occurred, but it's important to note the differences between natural philosophy and science as historiographical terms.
I want to end, however, with a quick talk about the very real anxiety the permeated thinkers all across Europe throughout the 16th and (mostly) 17th century. To say that era was a time of upheaval would be an understatement, and there was a grave existential uncertainty about the limits of human knowledge. To make a very long and complicated story short, scholars understood there to be natural forces, spiritual forces, natural illusions, and spiritual (including demonic) illusions. The problem was, there was no good way for us to separate illusion from reality, let alone natural illusions from those designed to confuse us. We can see this anxiety manifest itself, as I mentioned before, in a fixation on hidden forces, hidden symbolic meanings, and relational ties between mathematics and the physical realm (to say nothing of what philosophers like Descartes were up to). It's also one of the reasons that demonology really took off in this era; as crazy as it sounds now, demonology represented a concerted effort to understand specifically what demonic powers could do in the world and what they couldn't, and to build a foundation of certifiable and reliable knowledge from there. For more on that, though, I'd recommend Stuart Clark's Thinking With Demons, with the warning that it is by no means a quick or easy read.
I don't know if there are any specific questions that you had, but I hope this very, very general overview gives you some things to look out for. Nice question!
Sources:
Clark, Stuart. Thinking With Demons: The Idea of Witchcraft in Early Modern Europe
Moran, Bruce. Distilling Knowledge
Newman, William and Lawrence Principe. Alchemy Tried in the Fire: Starkey, Boyle, and the Fate of Helmontian Chymistry
Nummedal, Tara. Alchemy and Authority in the Holy Roman Empire
Shapin, Steven and Simon Schaffer, Leviathan and the Air-Pump: Hobbes, Boyle, and the Experimental Life
Shumaker, Wayne. Renaissance Curiosa
Smith, Pamela. The Business of Alchemy
[Edit: minor grammatical issues and one wrong century]