As a new player, what is the best way to maximize the energy we have available with our beasts. Especially at the start of a new day.

Quests > Explore > Battles ?

You're looking at two things: XP and silver.

The best way to accumulate silver is via questing, so try to adopt or buy beasts that can do one or more quests.

Exploring gives you both silver and XP, and is a slow way to level up pets. You do get food and materials though.

Arena battles give XP if you win, and are a quickerway to level up beasts.

Battlefield battles give you XP, and some sites give vermillion. The central portal gives top-level schemes and items every rollover in exchange for 100 of some material or other. You also can choose whether you want a wild symurgh or a throdama.

observation notes about energy use since i keep forgetting. (these could be wrong and i'll update)

Crafting requires 10 energy
Exploring requires 5 energy
Questing requires 5 energy
Breeding requires 1 energy