Is there a certain criteria I should look for to purchase pets for missons? Right now I need five mission pets, but Im not sure where to start for looking.

I went to Advanced Search and bought 5 high level sylesties, one for each stat, and made them super buff in one area. That way the stat is high enough that the mission doesn't fail

How did you do the last part? How do I make them buff in one area? Just equipment?

You know how the tutorial walks you through elements and a pet's proficiency points? Go to that page for each individual sylesti, and then choose an element and a stat to put their points into. My pets as examples:

Health -> https://www.sylestia.com/proficiency/?pet_id=6682174
Strength -> https://www.sylestia.com/proficiency/?pet_id=6710436
Intelligence -> https://www.sylestia.com/proficiency/?pet_id=6662859
Dexterity -> https://www.sylestia.com/proficiency/?pet_id=5863389
Agility -> https://www.sylestia.com/proficiency/?pet_id=6386795


I piled as many points as possible into one stat, so they wouldn't fail the missions. It's also worth noting that you have to check what element you attune a pet to if you want a pet to, say, have their points in strength over intelligence. Air, Earth, and Water are for the strength stat; Fire, Light, and Shadow are for the intelligence stat. Also, you have to have reached Umbra Forest to add points to dexterity and agility