The simplest way to get new cards is to buy them in the Shop. To begin, click on the Shop Tab.
When you are ready to buy, just click on an item and a little pop up will appear asking if you want to buy cards or not. Hit "OK" and the transaction will complete. Your new cards will appear to the right. The number of stars indicates the rarity, and the highest rarity cards will appear at the top of the list. If you click on them, you can take a look at their statistics and their flavor text. Hitting the "Skill/Text" button to the lower left will toggle between the flavor text and their Skill details.