The dragon-hunting quest begins when you defeat the dragon in Underground River. Much later, when you visit Kelaso village, the elder in there tells you that there are more dragons to be dealt with - that conversation makes them spawn in the game's world. Besides King Poo and Gold Mecha Robo, these are the most difficult enemies in the game (well, Moody Dragon isn't tough). Don't try to eliminate them too early - you really need to be well prepared for that.
Place: Underground River
Check out the Underground River section in DISC 2.
Place: southern outskirts of The Great Desert.
This dragon isn't impossible to beat, but it still has a whopping 25-30 thousand HP. It doesn't deal much damage, as long as you have decent wind protection. Also, don't ever attack it physically - it will heal it instead of dealing damage. Stick to the magic, folks.
Place: Coastal Road (south of Talta Village)
Get yourself some decent negative-status protection for this one (mostly Paralyze), as well as anti-Wind equipment. The enemy has lots of HP - more or less 15 thousand - so be careful and act wisely. Don't charge, even if that means you don't deal damage for two turns or more (heal instead). Remember that time is on your side - Corporeals are constantly loading up. When it comes to attack, try Windus and physical attacks.
Place: west of Kelaso Village
This monster mostly uses ice magic. To fight him you must have decent protection against it. The enemy is hellishly fast and deadly effective. In my case, I had to stick to healing and using Corporeals to kill it.
Place: Gul Mountains (behind a red barrier)
This one was the worst for me. Not sure if that was due to lack of proper protection, but it seemed like the most demanding of all the dragons. It required a little bit more training and better items. Remember to protect against wind and earth. Attack it physically or with fire.
When you defeat all the dragons (there's an achievement for that), get back to Kelaso and talk to the elder. He'll assemble a Dragon Bracelet for you - definitely the best bracelet in the game (you deserve it).