To start a culture you'll need a suitable container, clean water, food, and of course some specimens.
Getting your starting specimens
We all need to start somewhere. Daphnia and moina can be found in most bodies of water. You can even find them in some seasonal ponds or large puddles. The most sure-fire way to get hold of some though is to order them online. You can often find them in marketplaces such as eBay. Any species should be relatively easy to culture, but Daphnia Magna are a popular and hearty choice. They also range in size from small enough for fry to big enough to be enjoyed by adult fish up to about 6 inches long.
It's best to get at least 20 individuals. The more the better. Interestingly though, you need only a single specimen to start your culture. Because they spend most of their healthy adult lives reproducing asexually, it's possible to go from a single daphnia to an unlimited supply.
Starting your culture
Once you have some specimens you need a culture container. Any size container will work but it's best to have at least a liter of water to work with. For a more stable culture, use a bigger container. Commercial operations often use vats exceeding 100 gallons, but you only need enough space for a few individuals to live in.
They eat constantly so you should have some food on hand from the start. However, you can probably get away with just putting them in some dirty tank water from vacuuming your gravel or other fish poops at first.
Maintaining Established Cultures
Once you have a culture going, it's fairly easy to keep it going. The main thing to keep in mind is that you must harvest them, and can't over-feed them. Just keep feeding 1/5 to 1/2 of them to your fish, and waiting for them to re-populate.
Feeding your culture
Daphnia eat any free-floating organisms or nutritious debris that fits in their mouths. Their favorite foods seem to be green water and generally cloudy water filled with other microorganisms (such as from a dirty shrimp tank or fish tank).
It's best to culture daphnia alongside a culture of green water or other infusoria-rich cloudy water. They feed on suspended microorganisms such as algae and bacteria, as well as any nutritious debris small enough for them to consume. In a pinch, you can feed them yeast or even powdered milk. If you are jumping in un-prepared, you may need to rely on a powdered food source at first, but you're much less likely to experience a culture crash if you feed live foods to your live food. Green water is also easy to culture, and we have an article about that too. Click right here.
If you feed them powdered milk, yeast, or other "dead" foods, be very cautious not to let it go rancid. Start with much less than you think they need, and never add more if the water isn't clear within 24 hours. If it clears faster than 24 hours you can try adding more the next time. They will not likely starve, so be more worried about over-feeding than under-feeding.
Green water is the safest and easiest way to keep your culture going. You can add as much as you like, and if they don't eat it fast enough it's no problem because the algae will simply continue to live in the tank alongside the daphnia or moina.
No matter what you feed them, they need filtration and regular water changes just like fish or other aquarium pets. As they eat, they produce waste. They are such voracious eaters and reproducers that they can produce incredible ammonia spikes very quickly (and they can survive a surprisingly ammonia-rich environment for a short time). The best filter is a sponge filter driven by a water pump, because they do not like fast-moving water and can be killed by the impellers of a water pump.
Harvesting your culture
Harvesting is every bit as important to feeding if you want to maintain a healthy culture. In nature, they have limited food supplies and are eaten by just about anything that swims, so they usually don't end up in a "culture crash" because they are not being "cultured" - they are just a part of the circle of life. In your home, however, you can keep them well-fed and safe from predators, and they don't have any natural limits to reproduction before they create such bad water that they begin to reproduce sexually and lay eggs instead of live offspring. In short, harvest regularly!
It's easy. Use a net, or a turkey baster, or just dip a cup into the water to catch some. This last method is great because it doubles as a water change. Just wait until the water clears up, scoop some of it out and drop it in your fish tanks (happy fish time!) then replace the missing water with some green water or dechlorinated tap water. If you decide to harvest with a net, just be sure to also change the water periodically.
Any budding aquarist should have a test kit on hand. Test the water regularly. Water quality varies rapidly in these cultures. Ammonia spikes are particularly common because their populations can boom much more quickly than the bacteria that processes the ammonia and nitrites can keep up, leading to rapid fluctuations. The solution is to test frequently and keep their population in check by feeding them to your fish! If you detect an ammonia spike, simply perform a water change and slow down a little on feeding, or speed up a little on harvesting.
Keep a back-up
It's likely that something bad will eventually happen and you'll have a population crash. In most cases they will lay eggs in response to bad conditions, so if this happens you may be able to just leave the container alone for a few weeks and they could start to populate on their own. However, the best solution is to always have a back-up culture. The most effective solution is to have two or more primary cultures and harvest them in alternating order, then if one crashes you can re-populate from the other.
If you are short on space though, or only want one culture, it's easy to keep a back-up population in a jar or other small container. It's best to have at least a liter of volume, but just a few ounces can work. These creatures are hearty and can easily survive with very little food. They just won't reproduce as fast. Occasionally add a small amount of green water or even regular fish food or something to rot, and the water will have enough microorganisms to keep them going. It's actually much easer to prevent a population crash if you keep them kind of hungry, because predators are less likely to become an issue (see below) and water quality should remain pretty consistent.
Dealing with predators
If you are lucky your culture will remain healthy and keep booming without an infestation of predators, However, it is likely that you'll eventually get something like planarians or hydra in your culture, and these will experience their own amazing population boom because of how great a food source moina and daphnia can be. You can treat this problem with chemicals but the most reliable way to deal with this situation is to always have a back up.
It's probably easier to co-culture than to monoculture. All co-culturing means is that you have something else reproducing in there with them. Simple pond snails are a good option. They help keep the environment clean and produce some extra food for the daphnia, and they have even been known to eat hydra, which is great for daphnia. You can also culture them alongside shrimp, scuds, or just about anything else that won't eat them.