Let's get one thing straight here--Koi are carp. Common, basic carp; albeit dressed up for an emperor's party with their fantastic coloration. Carp in general are bottom feeders, as evinced by their drooping mouths and general omniverous tendencies. They will happily come to the surface of the pond for treats, and even become quite tame, however they are still carp. This means they are much more hardy than you would expect. They do need a large pond if there will be more than two or three, with a biological filter or other filtration methods in place; conversely they can also thrive in the average stagnant farm pond. Confused yet? Good. Just remember they are carp, despite their lineage.
Start with a typical garden pond scenario. A 10 x 10 spot is decided in the yard for the pond, or if the builders are feeling particularly ambitious, 12 x 15. Digging commences until they get tired or run out of money. This means the pond is quite fine as to surface area, but the depth is lacking. Depth is important in places with cold winters. If the average winter temperature causes the pond to freeze to a foot or more, exactly what are the fish supposed to do if the depth is only two feet? Surface area is important too; it indicates the oxygen transfer from the available water surface. Is there a specific formula for calculating surface area, pond depth, and filtration method? No, not really. Just remember they are carp and bear with me here.
I have experienced the sight of goldfish in an outdoor pond that froze solid along with their liquid medium. When the thaw hit, they immediately resumed life as it was. Koi are a bit more fussy. They need at least two times the depth of the average ice thickness. If the typical winter freeze in a specific area means ice 6-12" thick, then the pond depth should be 4-6', at least. Both goldfish and koi go into dormancy when the temperatures drop. As long as they have an area of clear water, they should be fine. Unless of course you happen to have cryongenic goldfish as I did, in which case you will wonder about the natural laws of the universe.
Filtration is a hellacious argument among fish owners, so do not discuss this at a dinner party composed of the aforesaid people. A small waterfall run by a pump through a biological filter works just fine in the average size pond. A biological filter has a stratum of round rocks, on which water flows through plants that love moisture, such as cattails, cannas, sagittaria, and acorus. The average biological filter should be at least a third of the size of the pond. Be prepared to maintain the filter. Cattails are fantastic for filtering the water, but they grow excessively. Nothing like trimming and uprooting them when the outside temperature is 35 degrees, the only time they are fully dormant. They can be thinned in the summer too, but do you really want to smell fish poo and algae? I didn't think so.
Feeding koi is actually quite simple. In the warmer months, feed the hogs as much as they will consume in twenty minutes. They are hogs, and will eat anything and everything. In summer, it is quite easy to tame them by giving special treats such as earthworms or commercial koi goodies; they can fussily decide between foods. Do not give them bread or other 'human' treats. Koi food should be a brand for outdoor fish, that enables good coloration. Winter feeding is even easier-Once the water temperature drops to 50 F, stop feeding.
If the koi are healthy, and the pond provides a good habitat, they will breed. Most of their offspring will be eaten, which is a good thing because you will be begging your friends to take a fish, dig a pond, get into water gardening, do anything to get rid of the sudden koi overpopulation that has suddenly afflicted you.