Simply level a dragon to 15 by feeding it as often as you can. Upgrading your farms allows you to produce food at different rates/amounts.I chose to upgrade all of my farms to level three, which gives you access to plant "Spring Cherry". That one gives you 7500 food and 8000xp. It's pricey at 200,000 coins but takes only one hour to produce.
As mentioned, you simply must feed your dragon enough to help it reach level 15. However, there is a crucial detail you must also be aware of, which is temples. Without a temple, a dragon can only reach level 10, and a temple for a corresponding element must exist in your land before the dragon can pass level 10. Note that for hybrid dragons, temples must exist for all of their elements. So for a Prairie dragon, for example, you must have an earth temple, fire temple, and wind temple built before it can level past 10.