Zeni is the game's basic currency or cash. It can be earned by completing missions, winning battles, Daily Rewards, or by picking them up in the battlefield map. It is used primarily in paying the costs for Training and Awakening.
Dragon Stones are much valuable commodity in the game since you can use it to perform "premium" summons, continue a game-over, restore your Stamina, and increase the your character slots. They're quite valuable as you can only get a single piece or a handful of them through rewards, completing maps and chapters, or from missions. As much as possible, save up your Dragon Stones to perform a Multi-Summon which guarantees a SR character on top of the 10 rare+ characters you'll draw.
You can purchase more Dragon Stones using real money but they don't come cheap well. If you want to keep the game a completely "free-to-play" experience, you'll have to be patient and work hard in earning those Dragon Stones.