My personal armory is well represented with pistols, rifles, shot guns and black powder guns, but no ARs. The obvious question, "What are you going to do with it?" is probably best answered, "Enjoy it. Maybe shoot a few coyotes or prairie dogs." I don't need/want a Close Quarter Gun but am very likely to use it in the 100 to as much as 300 yard range.
223/556, might even have a spot for 6.8
I'll take a stab at this. Take a look at the lower assembly tutorials already posted. Keep in mind that the AR-15 was developed as a military rifle. It's easy to field strip and clean, and even armorer level tasks are relatively simple and easy to do with minimal tools. All that to say, that the AR-15 type rifles really are user friendly.
.223 / 5.56 is a great way to start. Even with the panic and the still higher ammo prices .223 / 5.56 FMJ remains some of the least expensive center-fire ammo. I usually recommend a 1:7 twist bbl because it will handle anything but the lightest construction varmint bullets all the way up to the heaviest bullets that will fit an AR mag. The 1:7 twist is also great at handling the long for caliber copper solid bullets that are becoming more and more prevalent. However, you're thinking about shooting P-dogs, so you'd probably be better served with a 1:9 twist bbl for shooting those super light varmint bullets. The 1:9 twist will stabilize almost everything weighing up to 69-70 grains, and Hornady even says the 75 grain bullet used in their TAP lines will stabilize in a 1:9 twist bbl.
I know you don't have a desire for the in close uses of an AR-15, but at least take a look at some of the many barrier penetration threads that have been posted here. In short, ballistic tip and JSP bullets in .223 / 5.56 have much lower penetration than pistol or shotgun rounds. Just some good info for future reference.
You mention a combine, so I assume you're a farmer. When I worked in gun retail I sold many flat top AR-15 style carbines to farmers here in North Ga, and I always got positive feedback when these folks came back for future visits. Most of them mounted a red dot of some flavor on the rifle, and others used a fixed rear sight with the A2 style dual apertures. They all liked that in low light the red dot sight or the more open ghost ring style night aperture made it easy to get on target when a yote or other varmint needed to be dispatched. They liked that the flash hider kept them from being blinded in low light as well. They liked the light recoil and low muzzle rise that allowed for faster follow up shots (regardless of lighting) when there were multiple 'yotes to be shot at times like cows calving or a group of the things finding their chicken houses. The surprisingly good accuracy got the attention of all of these folks as well.
16" bbls and collapsible stocks were popular because they made handling going from truck to tractor to on foot very easy. The collapsible stock also allows the length to be adjusted in small increments to fit any family member or go from further out in the summer when wearing light clothes to a click or two in to account for thicker clothing in the winter. These farmers also liked the lower penetration risks mentioned above, and how their AR-15 carbines loaded with proper ammo didn't require any changes to go from 300 yard and in varmint buster by day to house gun at night. That's the near universal feed back I got from my customers who were farmers down here.
I like that by pushing out two detent retained pins on the lower I can remove the upper and install another upper literally in seconds. Since the lower is the serial numbered firearm, switching uppers on an AR is just like swapping bbls on a pump shotgun. That ease of switching calibers and / or configurations is one of the best things about an AR-15 type rifle.
Then there's the furniture. All it takes is removing and replacing one screw to switch out the pistol grip. Most collapsible stocks don't even require any tools to remove and replace on the buffer tube. Most fixed butt stocks only require removing and replacing a single screw as well. You can remove the hand guards by hand, but a $20 hand guard tool makes it easier. It only takes a minute or two to switch out hand guards either way. All of those parts are available in just about every shape, color, and texture you can imagine. It's very easy to make an AR-15 fit you physically and aesthetically.
As for other calibers, 6.8 SPC II is a neat caliber. It's almost a rimless extra short case 7-30 Waters equivalent, but it does require its own magazines. .300 BLK gets near .30-30 ballistics and uses standard .223 / 5.56 magazines. Those are two of the more popular calibers of many that will run on a standard AR-15 type lower.
I hope this explains some of the appeal and utility of AR-15 style rifles.