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In my world I have an advanced civilization whose technology is Magic based. Their civilization magic can be found everywhere though, it is found most commonly among the educated and Wealthy class, but even the most poor will be exposed to some type to Magic.
The only problem with this setup is that I feel that it robs magic of its mysticality, thus making magic just another form of Technology. Is there anything I can do to still hold on to the mystical feel of magic in my world?
Even if, Magic=technology, tech IS more mysterious than you may think.
So today, so many people use iPhones and computers. There's a huge amount of tech. But how many users actually know how it works?
They don't. They just know that it works.
It's mysterious to most people, except those who specialize in it. The difference is that we have the internet and can look things up.
Exposure to magic doesn't mean knowledge. So maybe there are ever-burning torches in every district, but your average poor person doesn't know or care how they work.
You can also delineate common magic vs. uncommon. Everyone accepts the things an average magewright can handle (like a light permanent light spell in a night club or the torches in the street) but anything beyond that is met with suspicion. Throw in some magical disasters related to the development of tech and a highly secretive order in charge of the development of magic for all.
One of the push/pulls I've looked at it is that magic is elite. Most people can't work it. So mages would want to make the lives of people better, just to stave off revolution.
You also may want to look at the advent of technology. Tech was/is frightening when it's next level. Early on people were afraid to toggle a light switch, because they had a fear of getting electrocuted. That's because they thought of electricity as lightning, which kills people, and they knew that lighting a room had something to do with that, but they didn't actually have an understanding of how it worked. In a few years they saw that they weren't harmed by turning on lights, and neither was anyone else. While they still didn't really understand, they accepted it as something in their lives.
Edit: Actually, the fear of being electrocuted in the early days of electricity wasn't totally unfounded. They hadn't yet figured out grounding, so you actually could die if things were not set up correctly or you touched a wire. Fires happened on a constant and regular basis because they did not yet have breakers in place. They used all sorts of stupid things for insulation like of all things PAPER. People would use a single light wire as the source for EVERYTHING in their home that was electric, so wires would trail down, and the circuits would get overloaded. If they touched it they would fry and everything would catch fire. Magic could be the same--as in, anything new and not completely researched might go on market.
Consider leaving your world's magic exactly as you've described it, robbed of its mystery and mystical nature by its common usage throughout your civilization. Then let your point of view character (and by association, the reader) learn that magic is still very much a mystery, and a dangerous one at that. Piers Anthony does this extremely well in the first few Xanth novels; showing magic as a pervasive aspect of his character's culture, then allowing one character to learn its hidden nature.
In addition to my answer to What's the smallest change to physics to allow magic, which suggests there may be more magic in today's world than we oft believe, there are some approaches you can take to to make your ubiquitous magic seem more magical and less scientific/technological.
One of the key aspects of science is the search for the so called "natural laws." These are aspects of reality which are true everywhere and at every time. Science tries to identify these laws, if they truly exist, so that we can act on them. However, there are known patterns which science is particularly poor at predicting. One of these classes of patterns is those which include the actions of the observer. In science, it is utterly forbidden for the observer of a scientific experiment to influence its effects once the experiment is underway. They are expected to set up the initial state and then let the experiment simply evolve according to the "natural laws." This is then used to distill the essence of those natural laws into equations we can use. If we permit the scientist running the experiment to interact with it, we run into all sorts of issues like confirmation bias that the fundamental scientific process simply cannot deal with. Scientists take great effort to make sure their science is not plagued by this with techniques such as double-blind testing and repeatable experiments.
So what if your magic didn't follow this rule? What if it was simply impossible to use magic without fundamentally altering the rules of magic in the first place? The alteration might very slight, but perhaps the cumulative sum of all of the magic use in your world causes the laws of magic to slowly change shape. If they did so, science would never find a "natural law" which never changes, so they would have to reject the hypothesis that such a law exists. However, if the changes were smooth, you would find that magic didn't change 'all that much' over the short run This would be enough for everyone in your society to use a particular magic, but find that it eventually fails them.
It would also open an interesting door for non-scientific approaches if magic changed differently in different areas. Perhaps, in one kingdom, there's one particular kind of magic which changes slowly within its extent. Outside of the kingdom, that kind of magic changes much more rapidly. This would create a unique flavor of magic for that kingdom because they would have more time to codify "scientifically" this magic's behavior. Another kingdom may have a completely different feeling.
I envision this having a feeling similar to surfing. We all know the ocean's surface is flat, but every now and then a wave comes by. As long as you ride where the wave wants you to go, you're surfing. And, from what I understand, its a feeling quite unlike anything else. You would have groups of individuals chasing magic waves around the world, riding them as they go.
However, some individuals would see deeper than that. While most will happily ride along on the waves created by millions of magic users around the world, others may realize that their actions have side effects -- especially around some key nodes where magic is unusually stationary. These individuals would plumb the part of magic which science could not touch: the kind which can only be had by entering that magic and experiencing it for yourself -- manipulating it and having it manipulate you. Magic may behave quasi-scientifically elsewhere, but near these nodes, the mages would rule the day, not the scientists.
You could include the fact that occasionally (only once in a great while) the magic will malfunction in completely unpredictable and drastic ways, and no scientist has a good hypothesis.
Similar to answer from User127202, but an explanation for why science can never explain it: Some types of cryptography are based on the data stream itself. In other words, if the last character transmitted was A, use table 1 to encode the next letter. If the last character was B, use table 2. This simple pattern can be reverse engineered, but it is possible to make an encoding that only can be broken if you know the entire data stream from the beginning.
Similarly, if you look at a fractal Lorenz attractor, knowing where you are on the graph tells you very little about where you're going next. You need the complete vector state to be able to trace the pattern. Even a slight roundoff error results in wildly different calculations. See also the Hugo-winner book "Three Body Problem".
If magic has laws that change over time on either a fractal or cryptographic basis, your scientists might be able to work out when the rules will change and even the range of possible changes, but because no one was around to record all the states from the dawn of time or because it is physically impossible to simultaneously measure every law to the absolute precision needed to compute ongoing states.
Given that, now you need a priest or oracle class whose job it is to announce when rules change (because they are constantly testing with tiny spells) and researching as fast as possible what the new conditions are.
We face this problem in cosmology: there is the potential for us to find a scientific principle that we can describe but never explain because the effect itself cuts us off from knowledge of the cause. There is some math work that suggests this MIGHT be true of dark energy, but nothing conclusive. So far, we've been lucky: our universe is generally both knowable and explainable, although weather prediction (distinct from climate prediction) is in the problem category.
PS: I must be getting old. Checked my library... I'd forgotten about Master Of Five Magics. It has one possible form of the kind of feedback-loop magic system I'm talking about: the rules change based on what people do with magic. https://en.m.wikipedia.org/wiki/Master_of_the_Five_Magics
The way people in ancient cultures thought of magic seems to be pretty much exactly what you want: Magic was common but mysterious, because it was essentially asking gods and spirits for favors. I'm basically suggesting what ohwilleke already said, but I think I can expand on it a little more.
Magic as contact with the spirit world replaces the laws of physics with the uncertainties of personal interaction - and personal interaction with powerful people whose type of power you don't understand. Imagine an old person asking their grandchild to "to see if you can do something with your computer to solve that". Here a mage is less of a technician and more of a skilled diplomat with good connections: He knows of the proper channels and the (very complex) proper etiquette to ask for something and he is in good standing with the other party - but he doesn't necessarily know how that something he asked for is achieved.
Also, this way everyone can have passing knowledge of magic, small reasonably reliable rituals for small favors. Imagine someone in a foreign country with a phrase book: You can go into a German bakery and say "ein Brötchen bitte" and get a bread roll, if you hand over some money (i.e. make a sacrifice) and it doesn't matter that you mispronounce the ö. But if you want to do something more complex like rent an apartment and make sure that you get a decent one (what would be a normal price? what do I have to look out for?) - then you'll have to employ a "mage", someone who speaks the language and understands the customs.
Of course for this to work you would have to design (or at least hint at) a decently strange and incomprehensible spirit world - ideally one that has something to gain from its interactions with humans, so that a civilization could be built upon common interaction.
easy, just make magic dependent on a mind. make a specific type of personality or mutually exclusive ways of thought one of the factors in making it work. if magic cell phones only work when nihilists build them, and magic railroads require a driver who is a go with the flow person, but the engine has to be built by a stubborn angry person. Then magical tech will be mysterious to most people even if they understand how they work.
Or maybe working magic requires a specific natural talent, but using a device they create does not, most people won't learn how they work because they can't do anything with the knowledge.
ohwilleke's idea of having magic tied to spirits or elementals that must be persuaded works well too, anyone can understand the machine but only a few people will know how to communicate with the thing that makes it work.
If you've ever read the "Twokinds" webcomic, conditions are very similar to those you describe. It uses some very sound principles of realistically writing magic into the fiction story.
The author makes a clear differentiation between good and bad magic, and how it can affect the person using it. It's clear that magic is a very complex art and while anyone has access to it, it comes more naturally to some people then others. That means that a lot of people can use simple spells but the people at the top are very knowledgeable and can use very powerful spells. As previously mentioned, showing that magic is dangerous adds some mystery to it, and the more powerful the magic, the greater the danger.
The other thing you need to do is not to let the reader know too much — if you explain how the magic works and in too great depth, it's boring — and the reader can start to guess the rest of the story. It can help to hint at great untold powers and other magical beings that have know all the magic there is to know etc.
I'd also recommend you read through some other examples for inspiration and ideas.
One of the big things about technology is that it is so damn predictable. If magic was facilitated, for example, by summoning demons or ghosts that had their own free will and weren't always easy to bend to your will, that would give it a completely different flavor than conventional technology. It could range from elf and the shoemaker type arrangements to powerful spirits with minds of their own that only powerful sorcerers could bind.
I play a Table top role playing game that has a bit of magic in it. The social stigma around it has only recently begun to lift allowing people to openly practice it in most cultures. There is still a lingering taboo about it though. Also, most people need to go through a difficult (sometimes deadly) procedure/ritual to obtain more power. This sometimes leads to side effects that can disfigure them. While it is genuinely useful to have the price to pay for a greater understanding is too high for many people. It is also a bit of a blunt weapon, not so much finesse, so people are very cautious when trying anything beyond the tried and true.
I would say, in short, you can make investigating it further a known danger with a high cost, both physical and social.
One way to get the mystery back is to make the magic non-generic: Namely that there isn't one unified magic (that all mages understand and practice), but several different kinds, in some sort of opposition/mutual exclusion. So (for example) energy-mages can't master life-magic and vice-versa.
Another tactic might be to make the magic's strength/effects variable, depending on things the mages don't understand -- or don't yet understand.
