After selecting the type of training, press quickly X. The monster will do the training by himself, and he will win if he's healthy.
Heart Crystal
When a monster dies he leves behind a heart crystal. feed this crystal to another monster to boost his skills.
Hidden Monsters
Disc - Type - Monster
Lunar: The Complete Silver Story (Disc 1) - Game - Desert Hengar
Lunar: The Complete Silver Story (Disc 2) - Game - Legendary monster
Monster Rancher 3 - Game - Moochi
Spyro: Year Of The Dragon - Game - Ice Gitan
Ace Combat 4 - Game - Desert Karakarakkung
Armored Core 2 - Game - Robot
Suikoden 2 - Game - Forest Raiden
Tekken Tag Tournament - Game - Wrestling Moochi
Fantasia 2000 - DVD - Joker
Magnolia - DVD - Frog
Men Of Honor - DVD - Merman
Bob Dylan: Greatest Hits Music - CD - Marin
Dave Grusin Music - CD - Koronit
Chrono Cross (Disc 1 or 2) - Game - Strange Fenril
Digimon World - Game - Forest Hare
Final Fantasy Games - Game - Fairies
Front Mission 3 - Game - Sea Octipee
Train: Drops Of Jupiter Music - CD - Robot train
Gungriffon blaze - Game - Moochi
Jokers
Train in Goat for awhile to get an item that will let you unlock Jokers from CD/DVD.
Skip animations
After selecting the type of training (for example, trampoline in Morx Forest), rapidly press X to skip the animations for the drill. It does nothing to the success or failure of the monster.