I have a Christmas card list and add names to it if we get a card from someone new. I never pack up the cards with my Christmas stuff....have been trying to store less and less.
I take one last sentimental look at the cards I received then throw them away. Except one card that was given to me by my niece and nephew three years ago (it has a picture of an adorable puppy wearing a Santa cap). It's so cute, I just can't part with it.
I have a small hard plastic bin about 14x14. I keep new cards, name labels, etc and cards I receive from current year. Next year I take out those cards so I know who to send to.
I put the Christmas cards I receive on living room mantel as part of the decor. I recycled them on New Year's Day except for four that had personal notes that I wanted to address next year.