Feeding your Dragons is super expensive and will distract you heavily from making Achaemenidae Gear for your Sub Generals. But it pays off in a lot of situations. For example, in rallies, your subs don't come along, but your Dragon does. Meaning you get more out of the Dragon in that situation than you would from any debuff gears you might have made. The same also applies when fighting over buildings in SvS, Battlefield, and Throne War. Most people feed their Dragons slowly over time so that they still have enough Materials to work on debuff gears.
At the top you see the EXP bar, showing your progress from one level to the next. In the middle, you have your current feeding options. These options randomize on each roll and can be up to 10 Red Materials. I pretty much feed the dam things whatever they want minus Refine Stones and Red Dragon Scales. You can only feed up to 30 times in a day across 10 rerolls of the feeding options.
Refining your Dragons is pretty simple. First off, never use the Badges option when refining, use Gems instead as Badges are too precious to waste here.
You will roll through refines until you get one within 5% of the max for that dragon for the troop type associated with the Dragon. Lock that and move on to getting another, rinse and repeat. As shown in the picture below, locking in a Dragon's refine doesn't cost gems like gear, instead it costs Dragon Crystals which are obtained from BoG. Each lock adds an additional 5 Dragon Crystals cost to the refine. Pasture Dragons cost 50 Refine Stones per roll whereas Wonder Dragons cost 100, both significantly cheaper than refining Ares or Achaemenidae Gear.