• Many ideas here (some of them are brilliant and very usefull) to keep them out of landfill: http://craftingagreenworld.com/2014/01/02/diy-crafts-33-ways-recycle-gift-cards-credit-cards-plastic-cards/
• Ideas for around the house here: https://www.thisoldhouse.com/ideas/10-uses-gift-cards 
• Many plastic gift cards are reloadable, so save it to reload over and over. Or reload and gift it to someone else.
• Or, better yet, give an electronic gift certificate, which can be emailed or texted with no physical materials being used.
• Some retailers have begun using biodegradable gift cards, simplifying the disposal process. And some will take gift cards and collect them for recycling. So when you buy a gift card, ask if that particular retailer has a recycling program. Others offer cards that can be recharged and reused over and over.

Bank, Credit and Membership cards with personal information embedded in them need to be disposed of differently. See how here: http://www.instructables.com/id/How-to-dispose-of-a-credit-card/?ALLSTEPS 

And remember to not put all the cut up pieces in one place at one time, spread them out so they can't be reconstructed for identity theft.

As a last resort, put them in the garbage.