Anubias, Crypts, Swords and most aquarium plants grow faster out of water. Besides floating plants that need to be on water, Vallis and Java Moss are the two main plants that need to be kept in water. Pretty much everything else can be grown out of water.
If you buy plants that have been grown out of water, you simply put them in pots with potting mix, and put the pots into a shallow container. Make up some liquid fertiliser (I use Thrive) and pour that into the container so there is an inch or so of liquid. Leave the plants in there and replace the fertiliser water every week. If the water gets used up during that time, replace it with fresh water that has no fertiliser in. You only fertilise them once a week and let them drink as much as they can.
If you are using plants that have been living underwater, you plant them up in a pot with potting mix and put them in a container with enough water to cover the pot by a couple of inches. The plant will either grow up and out of the water, or you can let the water slowly evaporate and as the level drops, the plant will develop terrestrial leaves. When it has plenty of terrestrial leaves, you treat it like a normal plant, as described above.
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Plants grown out of water will usually flower and you can collect seeds or take cuttings from them. The cuttings strike easily and you simply put them in a pot with potting mix and leave them in the container of shallow fertiliser and water. Within a few weeks they will develop roots and start growing.
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If you grow plants out of water, some of them will lose their leaves when put under water. Depending on the species, most will then grow new aquatic leaves and start growing.
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If you want to grow plants underwater, get some plastic storage containers and put the plants in pots with gravel. Put them in the storage containers and fill the container with fertiliser and water. The water usually goes green after a week or two but the plants keep growing. Replace the water each week with new fertiliser water.
When the plants get too long, take cuttings about 6 inches long and plant them in pots. They will take root and grow pretty quickly.
When you want to use these plants in your tanks, lift them out of container and let the water drain out of the pots. Put them in a bucket of freshwater without fertiliser. Lift them up and let them drain before putting them into a second bucket of fresh water without fertiliser. Lift them up and let them drain before putting them in the tank. This rinses any fertiliser out of the pots and off the plants.