As a short summary, energy is used regardless of the game mode you're playing as long as it is available. However, players need to understand that certain interactions require energy while some don't.
Mainly speaking, energy is needed to level up a player's Axies in Adventure Mode, as well as earn SLP in Arena Mode.
On the other hand, energy is not needed to earn SLP in Adventure Mode or gain trophies in Arena Mode.
The two tables below showcase what Energy is needed and not needed for.