In 5e, there's no separate psionics system. There are just some subclasses with a psionic theme. This is how they work in 5.0:
The Aberrant Soul sorcerer gets some extra spells that are either telepathic-ish in nature or creepy aberration stuff (arms of Hadar, hunger of Hadar, Evard's black tentacles, summon aberration). These can eventually be swapped out for divination/enchantment spells from the sorcerer, warlock, or wizard lists. The cost for casting these spells using sorcery points is lower, and when you do you do so without components (unless they have a cost).
More offensively, for those of us who don't like our psionics to be all Far Realmsy, at 14th level they get "Revelation in Flesh" which allows them to spend sorcery points to transform themselves in various aberrant ways, gaining various abilities. And at 18th, they can do some space-warping stuff for short-range teleports.
There's also the Great Old One warlock, which is basically a warlock with some aberration-flavored spells and telepathy.
The Psi Knight fighter basically gets telekinetic-flavored special maneuvers they can do, and the Soulknife can conjure "blades of psychic energy" which they can use in both melee and at medium range. They also have a small reservoir of psychic power they can use to boost skill checks, speak telepathically with a number of allies, and at higher levels do short-range teleportation, boost attack checks, and eventually stun people.
I really don't get why they are so intent on making the main psi-caster classes so tied to aberrations. I do not believe that's what most psionics fans want. Psionics fans likely want either Dark Sun, where psionics is super common and considered "clean", or Eberron where some psionics are aberrant-tied but others have connections to dream creatures, and in general have much more to do with crystals than with mucus and tentacles.