Daily Things to Do:
 stroke a horse in safe haven
 Defrost Frost   or My Frost for a chance to win a hypnos' blanket 
 congratulate up to 5 topaz for a chance to win a diamond or 50 equus
Topaz    Topaz   Topaz    Topaz    Topaz  My Topaz
 stroke up to 5 xanthos for a chance to get a horn of plenty
Mine Xanthos Xanthos Xanthos Xanthos Xanthos
Archimedes 
 Congratulate Someone on their Birthday!

SKILLS USED IN EACH TYPE OF COMPETITION:
Western:
Barrel Racing ~ Speed, Stamina, Gallop
Cutting ~Stamina, Dressage, Speed
Trail Class ~ Dressage, Trot, Jumping
Reining ~ Gallop, Dressage, Stamina
Western Pleasure ~ Trot, Stamina, Dressage
Classical:
Trot ~ Trot Speed Dressage
Gallop ~ Gallop Speed Dressage
Dressage ~ Dressage, Trot, Gallop
Cross Country ~ Stamina, Dressage, Jumping
Show Jumping ~ Jumping, Dressage, Speed

Links to search the sales for the companions: 
Sheep Hen Grizzly Hedgehog Butterfly Swallow Zebra Monkey  Llama Cat Squirrel Dog Panda Rabbit Goat Kangaroo Macaw Ladybug Bee Camel Owl Penguin snow monkey Giraffe Kangaroo Frog Turtle Bear Orca  Ostrich  Weta Akita Inu  
Griffin Phoenix Skoll &Hati Kyūbiko Namazu Ow Shika
 Chained Ghost Grimacing Ghost   Screaming Ghost
Axolotl dragon   blue    green   purple   red
Cat Siths: 1 2 3 4 5 6 7 8 
Monsters: (change last number to 2 - 28 for different versions) 

Link to pet bonuses