Dragons a naturally omnivores but will pick a limited diet fairly early on, which will be partly of choice and partly due to environment.

Dragons that are strictly carnivorous are often times on the smaller side to accommodate their prefered food source, smaller animals being the best chance of a meal for one while packs are needed for larger more dangerous prey. (Ice and Storm dragons are the most likely to be carnivorous.)

Herbivores dragons on the other hand have an abundance of supply, thus allowing them to grow to immense size, however they have to eat often due to the lower nutrient levels in most vegetation. (Fire and Earth dragons are more likely to feed on foliage.)

A carnivorous dragon suddenly eating like a herbivore or vice versa will result in sometimes serious digestive issues. Switching between requires patience and slow reintroduction, though there are many dragons that eat a balance of vegetation and meats.

Hatching to youngling can and will eat anything they can get their mouths around, including rocks, clothing and writing equipment.