There are many uses for pigeons. They lay eggs of course, so those are edible, although they do not just spit them out like chickens. They lay two to a clutch, and must be given time to recover and get some more calcium in them before laying again (if you take the first eggs).
There are utility breeds like Kings and Carneaux that grow big squabs, which are a delicacy in most countries.
Me? I race mine I love my racers (homing pigeons), and the sport is quite enjoyable.
Another thing you can do with homers, is the white dove release business. You need pure white homing pigeons, but they don't have to be racing or show quality by any means. They just have to be smart and have the homing ability to get home from the church.
You can also show any of the hundreds of recognized breeds.
Many people simply keep them as a backyard pleasure; no competition intended. But there a lot of neat breeds to choose from. Like Rollers, who execute several continuous backflips in flight, before straightening up and fluttering off again. There are also Parlor Tumblers/Rollers who roll in the same way, but on the ground. They'll tumble down your lawn for several feet/yards.
If you're only planning on getting one or two birds, they can easily be kept in something like an outdoor rabbit hutch. In fact, I have one now that I use as a breeding cage for one of my pairs.
If you'd like to have more than that, or allow your pair to raise babies (cause they gotta go somewhere, right?), then it's best to keep them in a loft. Basically a chicken coop with an aviary stuck on the front of it.
Here's a picture of my first loft, which used to be a chicken coop ( the small section where you enter, is still used for the chickens to get in and roost/lay eggs.)
Basically anything will work, as long as they have enough room, some perches, nest boxes for the pairs, good ventilation but not so much to where it gets drafty. Sounds a lot like what chickens need huh? The aviary added on the front lets them get some sun and fresh air, without it being on the ground where things can bother them or they might pick up parasites. They love aviaries, but it isn't required of course.
As for what to feed them, I feed a pigeon mix made by Brown's.
Pigeon pellets also work (chicken pellets are fine if no pigeon feed is available around you). Wild bird seed mixed with scratch feed would work as well. But what is really important is the grit. They need a mix of crushed oyster shell (for calcium) and crushed granite (for digestion aid) at least. We get red pigeon grit, which has all of that, plus charcoal bits for a healthy system, and coated in a pink supplement. Some people have red grit locally, others have a hard time finding it. But if you're close to someone with pigeons, chances are, they can help you out a lot with where to find what.
No matter what breed, pigeons will come home to roost, eat, drink, and lay, like chickens do when allowed to free range. Homing pigeons are the only exception IF they are too old to be re-settled to your loft. If you choose homing pigeons, get some under 5 months old. Birds 1-2 months old are the best age, since you'll only have to keep them as 'prisoners' for 2 or 3 weeks until you can let them out. Birds over 3 months, should be kept in for a month before releasing. And those over 5 months can be very iffy. It's best to play it safe in that situation, and use them as breeders, flying the babies.
Any other breed can also be released after 2 weeks of settling inside the loft, no matter what age they are. But with all flying breeds, it is best to get them as young as possible. Older birds will be "strong on the wing", meaning as soon as they come out, they're gonna want to fly a lot. In a new home, they may accidentally fly too much, too far, and get lost. But with younger birds, they won't be as flighty, and by the time they get flying good, they'll have their surrounding memorized so they won't get lost as easily
Most fancy/show breeds don't fly much, especially the big bulky feathered ones, or those with big tails (fantails) or big muffs on their feet. So any age is safe for them, since they won't fly a whole lot anyways (or well, not very far. They'll flutter about the yard though )!
One thing I would like to point out though. Hawks love pigeons. Even more than they love chickens. That's probably because pigeons are smaller than most chickens, and they fly, so therefore a larger variety of hawks can more easily catch them, plus they get the thrill of chasing them in the air.
With homing pigeons, all hawks will have some issues catching them, simply because they are so fast! But the smaller, woodland hawks like Coopers and Sharp-shins, are masters of sneaking up on pigeons and catching them off guard.
Fancy pigeons basically have no chance against a hawk normally.
So because of that, it isn't advised to let your birds loft fly (fly around the loft as they please) all day, but rather for maybe an hour in the morning, and an hour in the afternoon. The rest of the time, or any time you are not able to be outdoors keeping watch, they should be inside the loft where they're safe. It's best to feed them twice a day (morning, afternoon), and release the birds to exercise before you feed them. That way when it's time to come in, and they hear the food hit the pan, they'll come running because they're hungry. Makes it a whole lot easier to trap the birds in when they're on the hungry side.