Typically, most of it is just how you describe it. You can use spells with psionic qualities and just flavor them differently. You can decide if you want magic and psionics to interact or not. But mental spells (telepathy, telekinesis, Detect Thoughts, Calm Emotions, Charm/Dominate/Hold Person/Monster, even Bigby's Hand) are all good templates. Additionally, any illusion spell is also good to model mental assaults. Evocation spells too. You can keep the thematics, but just change any damage to psychic.
I don't know off the top of my head, but keith-baker.com is a great site where the main man himself discusses such things. There is likely a whole article about usinig the Quori and Inspired in 5e.
But when I run Quori or Inspired, they are above all subtle. They rarely engage in combat directly. Instead they manipulate those around them with patience and cunning to achieve their goals. They are all about subterfuge, misdirection, and sowing seeds of fear and doubt.