I'm going to split psionics up into three parts.
- Telekinesis (moving things with your mind)
- Telepathy (reading minds and speaking with your mind)
- Psychic viewing (sensing with something other than the normal 5 senses)
Telekinesis can be bound from a physics perspective: how much power does it take to make a pencil move, could the brain produce it, and could they do it without cooking themselves or the thing they're trying to move?
Let's get a lower bound. There's a scene in Phenomenon where George is idly making a pencil roll back and forth on a table with his mind. A pencil moving at walking speed has 6 milliJoules of energy, some more would be needed for acceleration but close enough. This is about the energy contained in a whisper or pressing 6 keys or a millionth of a food calorie. So yes, the human body can produce this amount of energy without cooking itself or the pencil or the table. We already know this because I can move a pencil with my finger, but it's good to get the numbers.
The next problem is how do you invisibly move an object using known physics? This is where the problems start. We could use a very focused magnetic beam, but it would effect everything in a line from the psychic to the pencil and beyond. Also pencils aren't very magnetic, so it would have to be extremely powerful to induce a magnetic field in graphite and wood likely cooking the user and everybody else. Finally, this would limit the psychic to only pushing things away from them or pulling them in. They couldn't, for example, lift anything, turn a doorknob or flip a switch up or down.
Maybe they ionize air to create a breeze or suction. Again, this requires a tremendous amount of energy, the breeze would be easily measured, and it could only push or pull.
I will leave the problem of telekinesis there. You're basically trying to make a tractor beam. Good luck.
On to telepathy. As has been pointed out by others, we can already influence other people's minds through the use of what I'm going to lump together as "social hacking". The human brain is a big pile of old parts repurposed and rewired together to cobble together sentience and intelligence (two ill-defined words), and it's version 1.0. There's lots of ways to trick it, cause people to think your idea was really their idea, or change their memory. Radiolab has an excellent episode on Memory And Forgetting about how every time we remember something we actually destroy and recreate that memory. They have another episode, Choice about how Free Will may be an illusion and your decision can be swayed by a simple cup of coffee. This is all a ripe field for your story.
As to mind reading, we need some way to transfer information from one brain to another. Here I'll take the term ESP (Extra Sensory Perception) literally. You can just sense things people can't.
I'll draw from the field of cryptology. It was recently claimed that encryption can be broken by listening to a computer as it decrypts something. Take that basic idea, stretch it a bit, and a mind reader may just be a person who can sense the minute changes in a brain's electromagnetic field and translate them into their thoughts. The brain is, after all, a supercomputer. Fish, dolphins and platypus can sense magnetic fields. Why not?
Finally, psychic viewing. Seeing through objects is simple enough, our psychic has the ability to see outside the visible spectrum. Biologically there's no problem with this, and some people can already do it a bit. Similarly, sensing very high and low frequency sounds is no problem. Adults tend to lose their ability to hear high frequency sounds, kids have ESP compared to adults. Assembling those sounds into "vision" (echolocation) is not a problem, humans already do it, your psychic just does it better.
As for things like taking a psychic trip to Jupiter (psychic or astral projection)... I got nothing.
There is another mechanism of action, that biological organisms can be Quantum Computers. Greg Egan has played with this idea. The novels are short and well written, so I encourage you to read them.
it's found that humans are the Quantum Observer and the main character has the ability to it turn off. He can try all possibilities simultaneously and select which version of the universe he collapses the decision tree to. Need to guess a pass code? Just randomly push buttons! One of the trillions of "you" will get it right, pick that one and toss out the rest.
DNA becomes a biological quantum computer. Evolutionary pressure, instead of coming from the current environment, is coming from all possible environments. DNA can predict the mutations necessary to survive future threats and reproduce.
A more subtle version of this, where the user was not entirely aware of the mechanism, would appear "psychic".
That's all about making psionics plausible. Will humans ever develop them? Evolutionarily speaking, you need to come up with a reason these chance mutations will help humans live to reproduce in the long term, and which existing part of the human anatomy can be adapted to fit. If we were trapped underground for many generations, rods and cones which can see a wider spectrum in dimmer light would be beneficial. If we lived in a cold climate, a mutation to continue to produce lactase after maturity allows us to gain caloric benefit from the abundant but otherwise inedible grass by digesting the breast milk of bovines. That all takes a long time, and is rather more complicated than is presented to the layman.
Fortunately, we have genetic engineering and don't have to wait around for evolution. Want to sense magnetic fields? Splice in some dolphin DNA! Of course, it's not that simple, but it's plausible. How do you do this to an adult human? Gene therapy! Alter a retrovirus to inject, modify or delete genes in your existing cells. Or use it to force the brain to grow new structures or make new connections. Still in the future, but plausible.