For the urban-surburban beginner--here are some ideas, cautions, reassurances, and so on. Some are the same as you'd find in a book or an authoritative source online, but some are my opinions. I'll try to remember to label those as "E's thoughts" but you will probably be able to tell anyway.
To get started, it really helps to have a very experienced person around to tell you what comes next. Even though you can find materials on the internet, when you're faced with the 3-D, full-color, full-sensory-experience reality, you are--the first time, at least--unsure of yourself, what your reactions will be. If you're not the one in charge, an unexpected negative reaction won't ruin anyone's day but yours. If you can find someone who home-slaughters and butchers on a regular basis and can just be helper a few times, that's ideal. In fact, helping out with other farm/ranch work--fixing fence, other animal work--can prepare the urbanized sensory array for some of the smells and sights ahead of time.
Since you're going to be eating the product of your labors, you want to be sure it's not going to make you sick. The first line of defense is choosing the right animal. You want healthy. If you're feeding the animal (rather than buying a live one for slaughter) remember that what goes into it is going to go into you. Never use seed grain for feed--seed grain is almost always treated with a fungicide, typically containing mercury. Yes, people in this country have had permanent neurological damage from eating meat from hogs fed mercury-treated corn. Never eat a sick animal. If you were going to slaughter a calf that looked fine two days ago, but is now staggering and bumping into fences...do not eat that calf. We don't have wolf or coyote digestive tracts; we need the healthy ones.
The next line of defense is cleanliness. How rigorous you have to be depends in part on how quickly you can work and how soon you're going to cook it. If you kill, drain, skin, and clean out an animal quickly and correctly, and immediately cook the meat, any *small* problems of cleanliness won't have time to multiply or penetrate before the fire gets 'em. After all, our ancestors didn't care that much about cleanliness (and some places don't today. But that's risky, and a bad habit to get into; most of us have citified digestive tracts. Better is to consider how to approximate the cleanliness of a good abbatoir. This means setting up a clean workspace and having a way to wash off the carcass after you've skinned and gutted it, to wash hands and arms and tools frequently and thoroughly. You'll need containers (the big heavy-plastic muck buckets are good) for the stuff you'll discard. Bleach is a good disinfectant for surfaces and instruments (but rinse with clean water, and it's wise to wear gloves unless you want chemically scorched fingertips.) A spray bottle of bleach is handiest for surfaces like work tables. You want your work tables to have an impervious surface. A pile of clean (!) old towels are handy for wiping hands at various points--one set for soiled hands (easier to get the hands clean in a wash bucket if you've wiped off some of the debris) and one for clean. You'll also need a roll or two (depends on size of critter) of paper towels for cleaning up as you go and a lot of plastic bags of various sizes and a marker or two for them.
The next line of defense is the right combination of quickness and care--both based on knowledge and experience. The quicker you get the carcass secured from possible contamination by insects and cooled down, the better--but at the same time you don't want it contaminated by the animal's own source of harmful bacteria--the gut. Necessary for both quickness and care are good tools--including extremely sharp knives. Dull knives plus slippery hands = knife-sticks....in you or your helpers or a part of the animal you didn't want to cut A boning knife (long sharp point) is handy for disarticulating joints; both that and a shorter blade are good for skinning (you can also buy skinning knives, but our ancestors managed with chunks of chipped flint...) A cleaver is another good choice. Knives and all other tools used should be perfectly clean (nothing clinging in the little crevice where the blade meets the handle, for instance.) Also useful is a saw (a hand-saw works fine on bone, but so does a butcher saw designed for the purpose or a stout hacksaw.) The saw, too, should be clean (and sharp, of course.) Electric saws and knives of various kinds are also useful. Have extra blades.
Looking ahead, you'll need a way to cool the meat before you do the actual cutting up--it's much easier to handle at 40F than body heat. Ideally (but we don't have ideal either) you have a nice big meat locker big enough to hang a beef half in until it's aged and you're ready to cut it up and freeze it. Less ideal is more-than-enough freezer and refrigerator space. Your packaging is likely to be less compact than a professional's would be, so a beast of the same size will take up more space. Also, you'll be putting warm meat into these cooling spaces and even a good freezer won't chill them to freezing that quickly. In six hours, the lamb we had halved chilled just to the good cutting temperature in an empty 9 cubic foot freezer. To help with that, you can pre-chill cuts in the refrigerator or in crushed ice before putting them in the freezer, and have the freezer loaded (at least half) with jugs of water that have frozen. Otherwise, every time you open the freezer to add another piece, you lose most of the coolth.
All this preplanning stuff may make you a bit queasy (it did me) but it pays off once you start: if you know how you're going to proceed (or your expert does) and you have all the equipment and supplies in place, it will go faster and more smoothly. That's better for both the animal and the humans.
Take another look at the space you're going to use. Where are you going to kill the animal? Drain it? What are you going to use to hoist it so you can skin and clean it easily (as easily as it gets for amateurs) if it's bigger than a rabbit or chicken? (People have used trees, engine hoists, tractor buckets--our way, jerry-built hoists of various kinds. Doesn't matter what you use, if it'll hold the weight while you work and not fall on your head.) Where are you going to dismember it? Where are you going to put the discarded parts temporarily? Permanently? If you're starting as a helper for someone else, look at how they organize their activity and think whether that will work for you or you want to arrange things differently when you're going solo.
Killing. Some people are good at the butchering but not at the killing (and vice versa, and some are good at both and some at neither. You can't know which group you're in until you're faced with it. E's opinion: since you can't know ahead of time, there's no sense beating yourself up if you fall into a group you wish you weren't in. If you've never done it before, you should have someone who has standing there to fix your mistake, if you make one. Cutting the throat kills quickly (with instant or near-instant loss of consciousness), but for larger animals requires more strength and the ability to make a deep, strong, wide cut in one swift movement. It's bad for the animal if you hesitate after a little cut, or need two cuts to make it all the way, etc. That's why my preference (purely my preference) is a single shot to the head, immediately followed by cutting the throat to bleed the animal. Even if you're an inch off, it will stun the animal (but don't be an inch off.) Immediate bleeding is necessary, as the heart stops beating and the blood starts clotting very shortly after the throat's cut. You want to get those hind legs up and let it bleeds out.
Blood. Which brings up blood. We've all seen it, but we may not have seen that much of it all at once. Remember--the animal's dead at this point. This isn't "injury" blood--it's not something you're supposed to fix (that was hard for me as a former paramedic, the first time I saw a lot of animal blood--the fix-it gene waved both arms.) It has a strong blood-smell, but it's not bad (for me, anyway.) It's an inappropriate smell for most of us--we're taught to cover up our organic smells--but there's no covering this one up when butchering. It helps to do this part outside, with plenty of fresh air around. A breeze is good, too.
Skinning is messy and can be difficult (despite the videos online showing pros doing it quickly and apparently easily--we're all over sixty and we use that as an excuse...those young whipper-snappers just yank on the hide and it comes off, seems like.) Ideally, you don't cut the hide or take meat with the hide. You want to get the hide off without the outside of the hide touching the meat. The outside of the hide has all the environmental dirt on it (and with a sheep critter, even sheared short, that's a lot.) With the lamb hanging from the bucket of the tractor, John could raise or lower it as needed for ease of work. You need a table nearby for the tools, towels, etc. And the hose and soap for hands. We do this on the gravel ramp to the barn.
With the skin off, it's time to clean out the innards. If you are careful, you'll get the digestive tract out without puncturing it and won't have the gut-smell to deal with. (If there's an accident, you'll need to wash down the carcass *copiously*--that's why the need to have a hose with good water pressure behind it.) It's easier if the animal was fasted overnight (which isn't long enough to cause distress) so the gut is fairly empty, but even if it's not, you can get it out properly , if you're careful with the skinning. Since I've observed in operating rooms ( during EMT and Paramedic training), I'm always interested in how healthy the innards of livestock look, compared to the humans who were having theirs worked on. Lungs are bright pink; etc. If you're going to use the liver, kidneys, etc., have their plastic bags ready to receive them--rinse them off, and put each in its own bag. Position the "gut bucket" under the critter, and make the *very careful* cut into the abdominal cavity so you can hold or tie off the far end before cutting it free on the outside...extend the cut down and let stuff fall into the bucket. And there you are. (Well, not in exhaustive detail--your expert guide will add more gritty and nitty.) If you haven't cut off the head and lower forelegs, this is the time to do that (the lower forelegs are really hard to skin, and are usually cut off with the skin still on--they're just bone and sinew.)
At this point we wash off the carcass, to remove anything that may've gotten on it from the hide or a careless touch.
If you have that handy meat locker, or a just-about-freezing cold vermin-proof barn (it's winter, say) you can now hang the carcass and let its own weight relax the muscle fibers as it goes through rigor mortis and cools. Otherwise, you'll want to halve it or quarter it and start cooling it with ice or refrigeration until it reaches a cutting temperature. For that, once you've halved it (cut down the midline--this can be done with various tools depending on your setup), you'll need to change the point of hoist and remove the still-skin-covered lower back legs (for the sheep, we just took down a half, laid it on a clean garbage sack on the table, and cut off the lower leg well above the skin.) We put these large chunks (half a lamb, less than that of a beef) in new plastic garbage bags to keep them clean as they chill and to protect the freezer from them as well. While the halves or quarters chill down, clean up all the tools you used and the table--toss those towels in the washing machine with cold water first (the ones that have bloodstains.) Do whatever you were going to do with the parts you aren't going to use, whatever those are besides digestive tract.
And now you have clean meat, ready to be cut up--or, if you're in a hurry, barbecue.