Gaomon
Level: Child
Japanese Attacks: Double Backhand, Gao Rush and Rolling Upper
English Attacks: Gao Rush/Speed Bag

Gaomon follows his "Master" Touma without question.

Gaogamon
Level: Adult
Attacks: Spiral Claw, Dash Double Claw

The evolved form of Gaomon with the Digi-Soul from Touma

Mach Gaogamon
Japanese Attacks: Winning Knuckle, Gaoga Tornado, Howling Cannon
English Attack: Gaoga Tornade
Level: Perfect

When MetalPhantomon is draining the life out of the DATS team Gaomon escapes and Touma breaks free with his soul he evolves Gaomon to Mach to defeat him.

Mirage Gaogamon
Attacks: Full Moon Blaster, Double Crescent Mirage, Gale Claw
Level: Ultimate

Touma evolves Gaomon via Digisoul Overdrive when the 3 Bio Digimon attack while he and the others are in training to control their Digisoul.

Mirage Gaogamon Burst Mode
Japanese Attack: Full Moon Meteor Impact
English Attack: Meteor Shackle
Level: Ultimate

In order to save Earth from RhodoKnightmon and the fact Rhodo nearly killed his sister Touma activates the Burst mode enabling Mirage Gaogamon to evolve.