While the process of activating arcane skills is like a matching game, activating mount skills is like playing rock paper scissors. As such, it will require you to be able to read your opponent (or current energy of eidolons if up against AIs) successfully.
Unlocking the Mount Skill System
Once the mount function has been unlocked, you will be able to use phase 1 and phase 2 skills, while the usage of phase 3 skills is unlocked at level 40. Setting/changing your equipped mount skills is done from the character interface. You can set which mount skills to equip by clicking on the icon listed as "1", "2", and "3" (once you hit level 40) or the green icons under them. You can only have a single phase 1, 2, and 3 skill equipped at any time.
All mount skills fall under 1 of 3 tiers: phase 1 skills, phase 2 skills, and phase 3 skills. The tier that a skill belongs to dictates how charged your Overpower gauge needs to be to use the mount skill.
Using mount skills require you to charge the Overpower gauge, which is done by selecting the energy that "beats" the energy that your opponent selects at the start of the round. Infernal beats Vital, Vital beats Glacial, and Glacial beats Infernal. Successfully "beating" your opponent's energy increases the Overpower gauge by half a bar and the words "Skill Recharge" will flash at the bottom of the screen.
The mount skill system can work against you if you are going up against another player that also has it unlocked, as picking an energy that is weak to the one your opponent picked will cause their Overpower gauge to charge. You'll see the text "Enemy Recharge" flash at the top of the screen should this happen.
Mount skills are automatically used once the Overpower gauge has been sufficiently charged. Capped at 3 bars, phase 1 skills require 1 bar, phase 2 skills require 2 bars, and phase 3 skills require all 3 bars. These skills also take precedence over all other actions, but in situations where two players both activate a mount skill at the same time (i.e. in team battle), then the player with the highest speed stat will use their mount skill first. Once a mount skill has been used, you won't be able to use it again in that fight until all available mount skills have been used. When that happens, both the Overpower gauge and all of your mount skills will be reset.
Every mount comes with 1 specific skill, which can be viewed by hovering your mouse over its icon in the mount interface. Once the mount has been unlocked, you will have access to that skill for as long as you have the mount. Should it be a timed mount, you will not be able to use the skill once it expires.
While not as easy to manipulate as the arcane system due to it boiling down to a guessing game, being able to read your opponent is still a necessary part of the fight. As with the arcane system, don't just focus on spamming mount skills; you need to be able to read the situation and plan ahead so you can use both the arcane and mount skill systems to their fullest. While the skills are good for a free hit(s)/heal/effect, it can be difficult to time if you're going up against another player real-time or if the AI-controlled eidolons you're going up against all have different elements.