After bloom care for orchids is little different than caring for orchids at other times. You should trim off the old flower stem, if it's turned brown. (If it hasn't turned brown, it may bloom again. This is especially common in Phalaenopsis.)
Otherwise, care for your orchid normally. Provide suitable temperatures for the orchid type, continue providing the amount of light it likes, and continue providing appropriate temperatures. With a few exceptions, keep watering your orchid normally, too. The most commonly-sold orchids will do fine in normal house conditions, if they get appropriate light and water.
A few types of orchids will need a dry rest after flowering, if they come from places with a dry season. (If that's the case, they typically bloom just before the "dry season.") A few examples of such orchids include Cycnoches, Catasetum, some Anguloas, Mormodes, some Oncidiums, some Dendrobiums, and a few Lycastes. When they are dormant, they are unusually susceptible to rot from overwatering, and don't need fertilizer. A good indicator that a plant is entering a dry rest is that it drops all its leaves. During a plant's dry season, don't water it. (If the pseudobulbs start shriveling a lot, water once every 2-3 weeks.) When it starts to grow again, you can resume normal watering.
After flowering is often a good time to repot orchids. They aren't spending lots of energy maintaining their flowers anymore, and many orchids start the next year's growth after flowering, so new roots can be expected shortly. So if the plant has outgrown its pot or its potting mix is getting old, consider repotting at this time.
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