Unlike the Psionic Class from XCOM: Enemy Unknown, the Psi Operative is not a subclass and instead has its own set of unique abilities. It can also learn all of the abilities on both sides of its perk table, if trained long enough.
The success of psi attacks is only affected by the operative's Psi stat - which is increased by using upgraded Psi Amps - and not Will. Will is only a factor when defending against psi attacks, and improving Will (such as through a Personal Combat Sim) will not increase the operative's Psi rating. In short: Psi = Offense, Will = Defense.
The psionic offense of a soldier increases with each ability the soldier learns, and with the level of the Psi Amp they have equipped.
Like the Will stat, Psi Offense grows at a rate of 4-13 per level with a cap of 100, however Psi Offense will always start on 50 at Initiate. This gives around a 50-50 chance of reaching the cap(avg 50+51 pts). To ensure maximum value, if you have the resources, discard or ignore acolytes with low Psi Offense stat rolls.
If you receive a high-level ability (eg: Fortress, Null Lance) as an option for training a low-level (Acolyte, Adept) Psi Operative, it will generally take longer to train that ability than if it is taken as an option at a higher level. If you're aiming to fully train your Psi Operatives, it will be quicker to choose training options commensurate with the Psi Operative's level (ie: those taking five days to learn with an Engineer present, ten without.)
Unlike other classes, the Psi Operative does not need to choose between abilities in their tree. Given enough time, a Psi Operative can learn every ability.
All Psi abilities (besides passive ones like Solace) require line of sight of the target to activate. This includes abilities that target allies, like Stasis Shield. All damage dealt by psionic attacks ignores armor.
Insanity is essentially equal to the Sectoid's Mindspin in its effects. It can cause Disoriented, Panic and, most importantly, Mind Control. Use it against particularly dangerous targets to potentially deny them of their abilities, their turn, or even turn them against their allies.
Keep in mind that if Insanity procs Mind Control, it will be temporary, only lasting 2-3 turns.
Schism upgrades Insanity in order to deal guaranteed damage to the target if it hits and will apply Ruptured to the target. The damage dealt will be improved based on the tier of the Psi Amp the Operative is using. This makes Insanity an overall useful tool against low Will targets, essentially marking them for death.
Stasis will remove all actions from an enemy, but makes them immune to all damage and prevents targeting them. It uses a single action and does not end your turn. Use it to deny enemy units their turn in order to protect your soldiers.
To maximize the potential of your soldiers, use Stasis to end the turn. This lets your soldiers deal damage to the target before you place it in Stasis, and when using Stasis with Stasis Shield on an ally, allows you to use their turns before placing them into Stasis.
Stasis removes Overwatch and Suppression if they are active on the unit, and also kills all Psi Zombies the target is controlling.
Stasis Shield upgrades Stasis so it will work on allies. This allows you to use it in order to protect your soldiers from all enemy damage. All Stasis effects apply here.
Sustain is a lifeline ability that causes a Psi Operative to be placed into Stasis with 1 health if they were to be killed. Sustain works a single time per mission. It is suggested to learn Sustain in order to protect your high ranking Psi Operatives from death.
Soulfire deals guarenteed damage to an organic target. The damage dealt scales with the tier of the Psi Amp the operative is using. Use Soulfire when aim chances are too low or damage is required immediately.
Soul Steal upgrades Soulfire to grant the Psi Operative 50% of the damage dealt from Soulfire back as health. This allows the Psi Operative to quickly heal themselves from minor damage if Medikits are not available or are being conserved.
Inspire grants the target an extra action. It is useful to be used on soldiers that have already expended all of their actions in a turn. Inspiring a Sharpshooter that has used one action point will allow them to fire their main weapon which is useful for repositioning or reloading. Additionally, it will not break concealment.
This ability also allows speed-running tactics to be implemented: Move up as close to the objective as you can get, move your Psi-Operatives together (3x3), cast inspire repeatedly on a soldier with a Wraith Suit and Superior Speed PCM, complete the objective, call in the Skyranger, and then extract your squad (this only works on certain mission types but can save a lot of time in the late-game where gaining XP & loot is a non-issue, such as Alien Facility Assaults).
Fuse is a useful, but situational ability that allows you to detonate grenades and other explosives on enemy targets, including dead enemies. Because it has no cooldown, it is a good alternative to other psi attacks because it deals instant damage in an area around the target. Because it also consumes their grenade, it is also useful against enemies who like to use their grenades often, such as Heavy MECs.
Domination is a staple ability of the Psi-Operative, allowing them to permanently Mind Control an organic unit. Domination has a single charge, and a 5 turn cooldown if the attack misses.
Multiple enemies can be mind-controlled at the same time if the Psi Operative successfully uses Domination and also gets this particular effect from Insanity. Note that if the Psi Operative is evacuated, all enemies under this particular Operative's mind control will break free on the same turn.
The unit you choose to Dominate can widely vary based on the situation and personal preference. There are certain advantages to controlling certain units, however;
Dominating an ADVENT Shieldbearer will allow you to use the Energy Shield for your own squad. This can be very useful for improving the longevity of your squad during a battle. They also have good armor and health values to make them good decoys as well.
Mutons and Andromedons are good candidates for Domination due to their massive health pool and very potent weaponry and melee capabilities, and their low Will makes them easy to Dominate. This makes them both a reliable decoy for your troops and deadly damage dealers against other aliens.
Codices have higher Will than most units, but their abilities make it worth it to Dominate them; While they won't use Clone and their weapon isn't very powerful, they can Teleport to instantly gain a flanking position on the enemy, and their Psionic Bomb ability can disable all enemies that use clips in the radius, along with dealing considerable damage if they remain in it.
Archons are fast, agile, and powerful units. While their high Will makes it harder to Dominate them, their melee capabilites, extreme vitality and dodge stat, and the Blazing Pinions ability can make them a good Domination target.
Sectoids are slightly more useful than the rest of these units, as they can Reanimate a single Psi Zombie, but their low health, melee vulnerability, their weak Beam Pistol and Psi Offense make them poor Domination targets.
Assuming you can bypass their high Will, Gatekeepers make amazing Domination targets. Their unrivaled health pool and armor values, their Consume ability to further improve their longevity, their Gateway ability to raise multiple Psi Zombie decoys, and the sheer power of the Gatekeeper Beam makes them one of the best Domination targets in the game.
Fortress renders the Psi Operative immune to damage from a Gatekeeper's beam weapon, grenades, Flamethrower/Hellfire Projector damage, all forms of acid damage (Acid Grenades/Bombs, an Andromedon's Acid Bomb), and all forms of poison damage (including Chryssalid Poison). This means that only melee attacks, psionic attacks, and standard attacks can harm the Psi Operative.
Solace is an ability that essentially grants a Mindshield to the Psi Operative and all nearby units. Consider removing any active Mindshields on your units and simply use the Psi Operative to cure debilitating mental effects. Units within the Solace field are immune to all mental effects, and a Psi Operative that moves to fit a unit with a mental effect will be cured.
Null Lance is an incredibly potent ability that deals massive psionic damage in a straight line that pierces all targets in a straight line. Null Lance can be used to attack even mechanical units and can be used similarly to the Plasma Blaster. The only real difference is that the Null Lance can be used multiple times per mission and does not shred armor.
Void Rift is a powerful AoE attack that deals damage to all targets in an area similar to the Rift ability from the previous game. Organic units within the radius have a chance to suffer Insanity. It is useful in order to apply Insanity to multiple targets that are close together.
Void Rift triggers Schism if it procs Insanity. Multiple enemies can be hit with Insanity from 1 Void Rift cast. You can get multiple Mind Controls from Void Rift. (Multiple MC: https://youtu.be/i1uf9xY5umA language warning.)