To trade for a draco pet you need a zodiac pet and the corresponding amount of draconis of the same element (10 draconis for newborn, 50 for adult, 250 for legendary). Arthomon (draco keeper NPC at varanas gates) will make the trade for you. A draco pet brings no benefits, but you do get it's monster card when you trade for it.
Draco trading options | draco card's bonus (newborn/ adult/ legendary):
Festival of fire: Capricorn pet + Geo draconis = Crius draco | +61/+91 /+152 defense
Flower festival: Aquarius pet + Aero draconis = Tethys draco | +28/+39/+73 damage
Masked Ball: Pisces pet + Hydro draconis = Oceanus draco | +61/+91/+152 magical defense
Spring rain festival: Aries pet + Pyro draconis = Coeus draco | +28/+39/+73 magical damage
Music festival: Taurus pet + Geo Draconis = Cronus draco | +61/+91 /+152 defense
Craft festival: Gemini pet + Aero draconis = Rhea draco | +28/+39 /+73 damage
Colorweave Festival: Cancer pet + Hydro draconis = Mnemosyne Draco | +61/+91 /+152 magical defense
Fairy Tale Event: Leo pet + Pyro draconis = Hyperion draco | +28/+39/+73 magical damage
Juice Festival: Virgo pet + Geo draconis = Iapetus draco | +61/ +91/ +152 defense
Pumpkin festival: Libra pet + Aero draconis = Theia Draco | +28/+39/+73 damage
Autumn Banquet: Scorpio pet + Hydro draconis = Themis Draco | +61/ +91/ +152 magical defense
Snowflake festival: Sagittarius pet + Pyro draconis = Phoebe draco | +28/+39/+73 magical damage
If you collect all 12 draco of the same stage (newborn/adult/legendary) you can show them to obtain a mount from Arthomon (draco keeper NPC at varanas gates).