I’m looking for a camera type which could provide some of the following functions:
In summary, my main needs out of a camera is maintaining high image quality of fast targets from a distance.
Your two requirements are orthogonally opposed to one another.
There's no such thing as "the best camera" for much of anything. They are all compromises between things such as speed, resolution, size/weight, and cost. All of them.
To get both higher resolution and faster processing requires better sensors with more pixels and larger/faster processors with more heat dissipating capacity. All of that costs money. So do very large lenses with long focal lengths and "fast" apertures.
Even then, one must choose between a very fast but relatively low resolution camera - such as the Canon EOS 1D X Mark II, Nikon D5, or Sony α9 - or a very high resolution but relatively slow camera such as the Canon EOS 5Ds/5Ds R, Nikon D850, or Sony α7R III.
I don’t have a dedicated camera. Just using an iPad, not good enough though. Distant shots are grainy, and fast-paced mages get blurred.
You have arrived at what you think is a solution to your problem (high pixel count, long distance shooting) and have asked about that. Instead, I'd encourage you to take your solution, shelve it, and instead ask the meta-questions: What resolution is necessary to take shots of some type (portraits, wildlife, etc), what focal lengths are needed for some type of shot (city, landscape, etc), and what types of cameras are available to meet my (now much more defined) needs?
You need to better understand your problem before trying to make a solution.
Distant shots are grainy
This is because the iPad/iPhone doesn't actually "zoom" to take shots. It has a fixed wide angle lens. When you tell it to zoom-in, what you're actually doing is telling it to crop the image and interpolate the result up to the normal resolution. Obviously, you can't create pixels out of thin air. Interpolation algorithms have become pretty darned good, but they have their breaking point where the image falls apart, appearing grainy/pixelated/unsharp/etc.
I'd actually encourage you to not bother zooming in at all. Simply take the shot and crop later - it's the same thing.
Your frame of reference for distance is also informed by your use of the iPad. It has a very wide angle lens - the type most use to capture a very large scene. So when you say that you need to capture something far away, you need to define that - as in, do you need to see someone's head and shoulders who is standing 20 yards away? Or do you need to see the whiskers on a cat that is 100 yards away?
The first of these examples is not that "zoomed-in" at all for the photographic world while the second is much, much more "zoomed-in." The first is much cheaper to obtain, while the second is out of reach for most people's budget.
fast-paced mages get blurred
Your iPad is akin to using a dedicated camera on full auto mode. While it can use really fast shutter speeds, it doesn't inherently know to do this. The limit with your iPad is the fact that the aperture is fixed. Normally, a camera can adjust exposure using aperture, shutter speed, and ISO but in the case of your iPad and its fixed aperture, you have only the ISO and shutter speed to play with.
Lower ISO's product less noisy, cleaner, sharper looking images. So, it's no wonder that your iPad is favoring the lowest ISO (thus slowest shutter speeds) for an exposure. Most people take selfies and shoot non-moving objects, so erring on this side means more good images for more people. Obviously, it's not true for fast moving objects.
You will see an immediate jump in your ability to shoot moving objects by downloading a camera application that lets you directly set the ISO and shutter speed so that you can force a faster shutter speed when you need it.
At some point, you will hit a wall and no longer get faster shots - this is where your ISO is maxed at its highest setting. The only solution here is to add light to the scene, if possible. You'll also suffer some image quality degradation at high ISOs, and if you want better high ISO performance, you will need a larger camera sensor (don't worry, every dedicated camera uses a sensor larger than the one in your phone).
About your solutions...
Resolution is less important than you think. Any dedicated camera made in the last 5 years will have plenty enough for you, unless your plan is to print images greater than 12"x18".
As above, focal length at great distance really, really needs to be nailed down. If it turns out that you need <200mm lenses...these are fairly cheap. If you need greater than that, well...they're not cheap at all. The best budget telephoto in the Canon line-up is the 400mm f/5.6, and its ~$1,000 all by itself.
Options that you have available to you are:
Before you make any hard decisions about what you need, I'd encourage you to better understand what is causing you problems, exactly the type of image you want to make, and then reach out to places like this for advice on solutions available. You don't know what you don't know, so ask.
I will probably not answer the question, but I hope to narrow your search first.
Provide high-resolution
High resolution is a relative term. There can be a "High megapixel" count blurry photo and you can have a sharper "Not that high megapixel count" one.
I would say that 24 Mpx image is good enough for almost any usage. Billboards, magazines, posters. And it a pretty common Megapixel count.
and detailed photos from long-range.
Some extra Megapixels can help you here, but you are on the realm of lenses. You need to define what lens you need. You probably need a telephoto. But your expectations are probably too board. A wild animal can be an elephant or a bird. You can be hiding or you can be on a Jeep...
Instead of that, think about a person, how far a person need to be to take a full-body photo of them?
Normally, the longer the lens the more expensive.
Can handle processing rapid-action images while still maintaining detail. Example, shooting auto racing.
Almost any camera can do this... If the camera can take pictures of about 1/1000 of a second you are fine. A normal entry-level camera can take pictures faster than that. 1/2000, 1/4000 etc... Any of those can potentially freeze a racing car... But we are not talking anything about how far you are from the car, what is the framing, what is the perspective, what are the light conditions... So, probably we are back to the lens itself, a faster lens will help you to freeze action better than a slow one.
In summary, my main needs out of a camera is maintaining high image quality of fast targets from a distance.
The moon travels about 2,288 miles or 3,682 km per hour, but you can safely take a shot with your phone.
The further away an object is the less apparent speed it has.
A moving hand in front of your camera moves way slower but it can be blurry. So, speed, blurriness, size of the subject, lens, megapixel, all need to be taken into account.
Some notes about the other answers and comments.
primarily high pixel and fast processor.
No. You do not need either of those necessarily.
Define your output size, print for example to define the Megapixels.
No. You can take a really good photo of a racing car near you if you have a good technique. On one shot.
Yes, an extremely high Frames per second can help you... But actually to know how to take photos will help you more.
My entry level crop sensor DSLR with a 75-300 f/5.6 lens, 3 frames per second, gives me fantastic formula one photos. Are they as good as professional shots? Of course not. Are they good enough that I'm proud of them? Yes, massively.
Don't be put off by the people on here saying you need to spend $10,000. To get really good, professional, photos you do, yes. But for an amateur who wants to get better photos than you can with an iPad, you absolutely can do it for something near your target.
My Canon 1300D (some as a Canon Rebel in the US I think?) with a 18-55mm lens and a 75-300mm lens cost me £400; you should be able to find a similar deal for $500-700.
There is something not mentioned in the question: the level of light.
If the action shooting happens during sunlight hours outdoors, an entry-level DSLR and a cheap crop telezoom will do what you want. The setup does in fact have some limitations:
For $500, I'm not sure if you can manage to buy everything (the camera, the memory card, the crop telezoom). For $700, you can.
However, if the action happens in artificial lights (indoors or during the night), you WILL need:
Note also that some lights flicker at the mains frequency (50 Hz, 60 Hz) or twice the mains frequency (100 Hz, 120 Hz). This depends on the exact technology used for the artificial lights. Thus, a camera which can take this into account and release the shutter at the perfect moment is useful.
Not only that, but you might find the 3 fps and small burst of cheap entry-level crop cameras limiting. Thus, you need faster burst rate and longer burst.
I have a Canon EOS RP that satisfies everything except fast burst rate. I wouldn't recommend it for primarily action. For primarily action, you should be looking at Sony's full frame mirrorless cameras.
If going the expensive route (fast lens, full frame sensor, high burst rate, long burst), you should put a zero after the planned investment, i.e. $500 becomes $5000.
My recommendation would be to go with the cheap < $700 equipment first, IF the shooting happens in sunlight situations. Then you will start to learn about the limitations of your equipment and are better equipped to search for eventual replacement.
However, if shooting in low light, there is no way around needing a fast lens and preferably a full frame sensor that is fairly modern (although some could say 70-200 / 2.8 with a good crop body could offer a decent range of focal lengths at fast aperture, but f/2.8 on crop is equivalent to f/4.5 on full frame so you're not that far below the aperture of a 100-400, and 400 at full frame is longer than 200 at crop).
