What is the best way to print cards for my game?
You could use common card protectors like people use to protect cards from trading cards games (Magic, etc). What your stuff in there (printed cardboard, paper-glued-on-other-cards,...) then becomes pretty much irrelevant.  

If not, what about photo printing services? Look for ones making postcards, you can probably easily fit 4 cards on a potcards and they ought to be rigid enough.