Here are my 2 cents:
For the stated purpose of taking care of kitchen or simple knives as used at home, garden etc, your options are: 1) Natural 2) Synthetics 3) Diamonds
Naturals are divided broadly water stones and oil stones.
Water stones: Japanese Kyoto stones such as Nakayamas, Uzukis etc, Belgian coticules, German Thüringen's, Chinese Guanxi etc.. are some of the most common ones, but almost every country has some found sources of water stones as used by butchers. As far as oil stones, the very best ones are Arkansas stones, made of Novaculite, mined right here in the good ol USA, and can be bought online or found in auctions for cheap. People used to use them widely in the past, so they can be found in many garage sales.
Synthetics and diamonds are also options, and for many this is all they use. While they are efficient, they generally cut too fast, as they are considered aggressive cutters, resulting in edges that are over sharpened and may end up loosing their sharpness too soon. Beginners like them as they are easy to use, but they are known to grind the edge too much and too fast.
Out of these options, here is my recommendation: as the first and only purchase, your best bet in my opinion is a set of good natural Arkansas stones, typically a soft (roughly 700-800 grit) and hard (roughly 1000- 1200 grit) Arkansas, and maybe a translucent and/or a black surgical (very high grit polisher, used in the past to sharpen surgical equipment) and you are set for life. These are very special stones. As they are not aggressive cutters, they are the best polishers, and they last forever for many generations. A Norton crystolon (synthetic) which is low grit (280-300 grit) , is also a very cheap and a good option for setting the primary edge (the bevel). The edge profiles from Arkansas stones are very keen and very long lasting. It is hard to over sharpen with oils stones. Over 100 years ago they were the most expensive stones in Europe where barbers used them for sharpening razors and surgical equipment, and considered them very special. This is in times when hand sharpening was a very common and widely used skill in many trades. A typical 6-7 by 2-2.5 inches stone is all you need. Ideally get them in 1 inch thickness, as that way they will last for generations. As mentioned above, a three-stone set of Washita Oilstones are a great option, and can be found online or on auctions.
The type of steel you have is important to know when selecting the stones. For normal carbon knives and German stainless steels which is what most common cheap knifes are (and cheap Chinese steels), Arkansas stones are perfect. You may use oil , but you can easily use plain water with two drops of dish soap (Dawn) to increase surface tension, and you do not mess your knives with oil. If you have the newer alloys, like Japanese VG10 and above, or powders, you may need synthetics, or diamonds as they are hard to sharpen and need aggressive stones.
A good diy leather and canvas strop is also a great cheap addition with great benefits (YouTube is your friend). Do not ever use sharpening steels, as they damage the edge. There are angle assisting devices available, usually expensive, and mostly using diamond or synthetic stones. I find them redundant, but they can be useful for the initial setting of the bevel, an important step.
Last, but not least: Japanese stones (known as jnats) are a very special category of water natural stones, primarily originating from Kyoto. They are more expensive, but very much worth it, as they produce very keen and smooth edges, even in hard alloys like Japanese ones. They are used with a rubbing stone (called a nagura) and typically you might need to use two-three different naguras in progression from rough to finer grit, and as you rub them on the stone and dilute with water, they disintegrate into very fine particles producing very fine edges. If you want to be amazed, check: TomoNagura.com (not marketing my site by the way), simply directing you to some cool info I have found in the past. Having them for kitchen knifes is a luxury. They are suited for real passionate connoisseurs or chefs who use and need high end knifes that need to be very sharp, such Japanese or Asian fusion restaurants that need to cut fish decoratively. For most everyday kitchen use, they are an overkill.
Selecting stones goes hand and hand with selecting knifes. For home use, I do recommend softer steel knifes like German steels (Henkels, Victorinox etc) or some of the better Chinese steels as they are very forgiving, durable and will not chip, and are easily sharpened quite keenly with Arkansas stones. A good place to look for them are restaurant supply stores, as they have steels and designs that can endure heavy restaurant use.
Japanese high end steels. or other high end knifes with hard alloys will produce extremely sharp edges that are very fragile and will chip. Unless you are into sushi, there is no need to go after them, but yeah, they are special! If for some reason you buy synthetic stones, the Japanese synthetics are in a class of their own, absolutely the best. Get what you can afford, but they are far superior than the rest of synthetics in both durability and function.
One more thing: it is important to learn about getting your stones flat, and how to achieve that with your particular stones (again, YouTube is your friend).
I sincerely hope you find this info useful, as I see so much confusion online. Our frugal predecessors were masters in freehand sharpening using natural stones, with no permies, no Youtube, no TikTok, no Facebook and no phones! As much as we have devolved, I think we still can do it!