I might be missing something (probably am) but I cannot find a list anywhere that tells how much energy points are taken by the various activities - how many points to explore, do a quest, be pregnant, etc. I suppose I could figure it out by trial and error, but that seems hard.  Anyone know?

Exploring: you lose 5 energy when you go through the portal, except if you start with exactly 5 energy (then you lose 4). 

Breeding: both parents lose 1 energy point. 

Quest, craft: when it's done, the pet's energy is reduced to 0. 

Losing a battle (on arena or explore) costs 1 energy. A draw in the Arena also costs one energy, but in explore draws don't cost energy at all. 

Every time you move or attack on the Battlefield, all pets on the squad lose 10 energy. 

Thank you!! Is there a minimum amount of energy you need to go on a quest/craft?  Or can you send a pet on a quest with only 1 energy?

There's a minimum for both--5 for quests, 10 for crafting. (You can check by hovering your mouse over the question mark by the page title--I know I always forget which of those has which requirement.

great!  Thanks for the information, that was exactly what I needed to know.