Well to make it simple, there are only two abilities;
-psychic abilities that are just that, psychic.
-psychic abilities that manifest themselves in a physical manner (telekinesis).
The first consists of many different abilities, like telepathy, empathy, precognition, psychometry, clairvoyance etc.
The second is telekinesis/psychokinesis. Forget about all the stuff that you hear about cryokinesis, pyrokinesis, electrokinesis, airokinesis, poopykinesis and whatever else. They are all the same thing (you affect the physical world with your mind. This can be objects, molecules or air currents or whatever). The only difference you might want to make is micro tk and macro tk. Micro tk affects chance, like how many times you will get a 6 when you throw dice. Macro tk is moving objects and stuff.
And how to train this stuff...... I wouldn't know. I'm not able to do any of those things myself. If I was, then you would probably see me on tv (unlike all the fakers that claim to have powers but do not want to prove them)

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