how do you take care of your hands after rock climbing?

I usually put a vitamin E cream on them.
There are creams and balms that are made specifically for climbing, but I have always benefited from pure vitamin E. I haven't tried any of the others climbing specific creams or balms, though.

I have a cream that is very heavy in vitamin E, I put it on either the same day after climbing if I remember, or a day later, and it recovers my skin pretty quickly. After an intense bouldering session, my skin can be left pretty thin and sensitive, after I put the vitamin E cream on though, it feels much better and is ready for another session.

There are also vitamin E oils, which may have better results, but are messier. They are probably best applied at night before going to sleep, that way you don't touch anything with the oil. Another option is to put on finger gloves that cover just each finger, this way the oil doesn't touch anything.