I don't throw anything away that I might be able to use and greeting cards are great for my grandchildren to cut out or to make cards from. Periodically, I go through all my paper, card and old greeting cards and make new ones. (Paint chips cut into shapes are a great source for added color).
You can purchase ready cut card, but the cheapest way to make them is to buy a 8 1/2 x 11 pack of card stock (I buy white) and cut each sheet in half (8 1/2 x 5 1/2), then fold those in half. The finished card will fit into white envelopes that you can find easily or you can make your own (more on that later).
Look online for ideas for your cards (Pinterest has good suggestions).
You can usually purchase the Happy Birthday stickers at Michaels or other craft stores (I've found them at dollar stores too).