What is the correct way to store a bow after use? Is there any specific care that is needed pre and post storage?
It all depends on the bow. I'll run through the three most common.
For a recurve.
Never left strung and detach the limbs. My personal superstition is make sure the top and bottom bolts always go in the same locations, but this isn't proven to make any difference. Make sure it is kept in a dry place. You can get some nice padded boxes to store them in. When restringing make sure it is twisted enough and not frayed, and give it a rub with some solid beeswax.
Longbows are similar.
Always unstring. Keep in a dry location. Also make sure it is stored on a flat surface to prevent warping. You can get some nice cases to store these in as well. Same goes with the string as the recurve; once strung give it a good wax.
Compounds are different.
Leave these strung. Some people ease off the tension before putting away, some don't, it seems to be personal preference. Once again store in a case. Before using, waxing the string is also a good plan.
From what I have noticed many people have their own answers or rituals that they swear by; so I am sure some answers other than my own will crop up as well.