The way I’ve heard it is that you can safely fire .223 in a 5.56, but not the other way around.
Ultimately the consolidated answer to the OP’s question is that caliber is the diameter of the bullet. That can be measured in metric or US standard, or sometimes both. For example, .357 inches = 9mm, 7.62mm = .30, and so forth.
The metric method is generally written as bullet diameter x case length. So you’ll see things like 7.62x39 (AK-47 round), or 9x19 (the classic 9mm pistol round), or 7.62x51 (7.62 NATO). Sometimes there’s a further descriptor - like 9x18 Makarov. The US standard system is typically the diameter and the manufacturer or some other descriptor. So you have .308 Winchester, .375 H&H, .38 Special, .44 Magnum, etc…
Sometimes these measurements aren’t quite accurate. For example, the 9x18 Makarov is actually 9.27mm, instead of the exact 9mm of the 9x19 Parabellum. Or that .38 Special is actually .357 (which is also 9mm, FWIW).
Shotgun shells are their own thing- the “gauge” indicates the number of lead balls of the bore diameter that adds up to a pound. So a 12 gauge has a diameter of a lead ball 1/12 of a lb. They also come in lengths - 12 gauge shells come in 2.75", 3" and 3.5" lengths for example. So you might see a 3" 12 gauge shell. The shot (the small balls) in the shotgun is described by number- #9 shot is very small, and #1 is fairly large. Then you get BB size (just like a BB gun fires), and up to T sized shot. Then you get into buckshot, which is the same sort of idea, but different sizes, starting with No 4, which is the smallest, and all the way up to 000 buckshot, which is the largest.
Magnum generally means a higher pressure (more gunpowder) version of the base round. So you’ve got 38 special and 357 magnum, where the 357 is the same bullet, and the same case but just a bit longer so it can’t be fired in a 38 special gun. Or 44 Special and 44 magnum.
Rimfire is an older priming method, where the cartridge case is manufactured with a hollow rim, and primer compound is put in that hollow rim. That way, the firing pin just impacts the rim and fires the bullet. This is opposed to centerfire, where a separately manufactured primer is pressed into a little hollow in the center of the cartridge case. Rimfire is typically used in small-bore cartridges- specifically the .22 long rifle, 22 short, and 22 magnum cartridges these days.
You’ll see bullet weight as a secondary descriptor on many cartridges. It’s almost always measured in “grains”, one of which is roughly .065 grams. So you’ll see 124gn 9mm rounds, or 55gn .223 Remington rounds, etc…
Full Metal jacket, hollowpoint, softpoint, etc… are different types of bullets. Full metal jacket (FMJ) means that the entire bullet, save maybe the base is encased in a copper jacket. These typically don’t deform on impact, or at least not much. Hollowpoint is what it sounds like- a bullet with a definite cavity in the nose intended to cause the bullet to deform in a specific way. These are usually low-speed bullets primarily used in pistols. Softpoint is the same idea, but for high powered rifle rounds. At those speeds, they don’t need a cavity because a soft exposed lead point ahead of the copper jacket will cause those bullets to deform into a mushroom shape, just like the hollowpoints do at lower speed. In fact, IIRC hollowpoints at high speed will actually fragment more often than softpoints.
Powder volume is dependent on what sort of powder is in the case, and you can find that out from reloading manuals. It’s generally not important for buying ammunition.