Which of these three terms is the most relevant in a resume?
Should any be avoided?
For clarity, I do understand the irony of pretending to be a self learner posting questions on StackExchange, thank you.
self-taught: I would use this with reference to a specific subject/technology/whatever which I had taught myself: I am proficient at X (BMgc, Unseen University), Y (Dip, University of Krull), and Z (self-taught) or I taught myself Z or I am a self-taught expert at Z.
self-learner: I probably not use this word, but if I did, I'd use it to describe a general temperament, rather than a specific piece of self-teaching: I am a self-learner. (It would be better, probably, to say that you were a good learner, and self-motivated (or, in the language of résumés, a self-starter).)
autodidact: I like this word, but, depending on what you're applying for, it might be too hifalutin to go on a CV.
Hence literary sources suggests that 'self-learner' isn't been used that very often compared to the others.
But in cases where you want to suggest that you have a quality to take initiatives and learn things on your own (like in the future), 'self learner' would fit fine. And for times when you want to suggest that you have already acquired a particular skills or knowledge I think, 'self-taught' would fit better pointing towards those already acquired skills.
The question of which word will best serve your goals is probably unanswerable here. It depends on the mood and assumptions of the person who first sees your resume, more than anything.
While I generally agree with the other answers here, I think one key point of the difference in meaning is being lost.
"Self-taught" carries connotations that "autodidact" avoids. When people hear "self-taught" they often assume a process like learning bits and pieces to help you accomplish whatever goals you were working towards at the time. They'll often assume it means you lack the deep, fundamental understanding that comes with a formal education or experience.
"Autodidact", if a person understands the word, seems to carry a connotation of broader, more thorough "self-education" rather than simply picking up a skill that wasn't formally taught to you. This is someone who doesn't just teach themselves skills, but investigates a broad range of topics to the same depth one might experience in formal education.
So as TRiG says, "Self-taught" should be more about some specific skill you picked up rather than a broad range of study.
If you are trying to communicate "autodidact" in an area where you may be misunderstood, try "self-educated" instead. And maybe elaborate on your meaning with a paragraph about that in your cover letter, or a personal summary section.
Auto-didact sounds, to many of the lesser educated types that screen your resume, very ivy league pretentious, in a snobby way.
Point out your skillset and point to your education where there is a mesh, and indicate that you extended your learning to develop and extend skill. Self taught with regular peer review to develop new skills and stay at the cutting edge.