I can't imagine that I personally know anyone who has the faintest idea what they would need to study in order to be "learning the mysteries". I do not think there is any such book or verse. Where would we look them up at? What page would we turn to? None will suffice. They are, apparently, mysteries.
But here is what we are told to do, mystery or no, we are commanded to ask, seek, and knock. We are given the gift of the Holy Ghost. It is not without purpose that the Lord makes us this offer:
Moses 6:61 Therefore it is given to abide in you; the record of heaven; the Comforter; the peaceable things of immortal glory; the truth of all things; that which quickeneth all things, which maketh alive all things; that which knoweth all things, and hath all power according to wisdom, mercy, truth, justice, and judgment.
In fact, the Lord makes clear the meaning of offering us the gift of the Holy Ghost so explicitly we cannot misunderstand:
Moroni 10:5 And by the power of the Holy Ghost ye may know the truth of all things.
And that is not learning the mysteries. That is just learning. That is necessary from square one, for can we gain a testimony except by revelation?
When Brigham Young and others have told people to stay away from the mysteries, they have been giving precisely the right counsel. They prevented people from wasting their time and becoming fascinated with trivial matters that are of no moment, deluding themselves into believing they had happened upon the mysteries of God simply because the matter was too obscure or odd or simply wrong for anyone else to know much about it. Such attempts never were and never are founded in the Holy Ghost. They kill the spirit, frequently replacing it with an academic fascination with matters of no spiritual moment whatsoever. They are turning aside from the type of learning whereby those members gained a testimony - instead of learning by the spirit of God, they are seeking some other spirit or method instead. And that is something the prophets have rightly opposed.
Such deluded attempts to delve into mysteries are a tool of the adversary that plays a role very much like his false manifestations. When someone who is noticeably spiritually off kilter tells, or hints to people that he is studying the mysteries then the very idea of the mysteries of the kingdom becomes tainted in the minds of other members who then want nothing to do with them. It is precisely the same method Satan uses to turn people cold to dreams and visions. He inspires someone who is easily deluded into dreams and visions that do not come from God, gets him to hint about that to others, or even to openly publish them to the world. Next thing you know the saints of God themselves have no interest in the dreams and visions of heaven.
Thus Satan uses the easily mislead to keep the flock at large from accepting the plain teachings of the scriptures regarding mysteries, dreams, visions, and revelation. And without faith, God can do no miracle among them, for faith ALWAYS precedes the miracle.
By contrast, God mostly use the word "mysteries" in the scriptures to mean those things that are hidden up because of unbelief, not those things that are trivial and of no moment. Can those things that are hidden up because of unbelief be learned except by belief? Can that which can only be learned through the Holy Ghost by belief be harmful?
Jacob 4:8 Behold,, great and marvelous are the works of the Lord. How unsearchabel are the depths of the mysteries of him; and it is impossible that man should find out all his ways. And no man knoweth of his ways save it be revealed unto him; wherefore, brethren, despise not the revelations of God.
Moroni 7:16 For behold, the Spirit of Chirst is given to every man, that he may know good from evil; wherefore, I show unto you the way to judge; for every thing which inviteth to do good, and to persuade to believe in Christ, is sent forth by the power and gift of Christ; wherefore ye may know with a perfect knowledge it is of God.
17 But whatsoever things persuadeth men to do evil, and believe not in Christ, and deny him, and serve not God, then ye may know with a perfect knowledge it is of the devil; for after this manner doth the devil work, for he persuadeth no man to do good, no not one; neither do his angels; neither do they who subject themselves unto him.
D&C 42:61 If thou shalt ask, thou shalt receive revelation upon revelation, knowledge upon knowledge, that thou mayest know the mysteries and peaceable things - that which bringeth joy, that which bringeth life eternal.
And they are founded in learning by the light of Christ and by the Holy Ghost. No, I make no claim to knowing the mysteries spoken of. But I do claim that the scriptures pertaining to them are perfectly true, and that they are founded in learning to seek revelation by the power of the Holy Ghost.
The scriptural discussion of the mysteries is quite plain. The doctrine is this:
Alma 12:9 And now Alma began to expound these things unto him, saying: It is given unto many to know the mysteries of God; nevertheless they are laid under a strict command that they shall not impart only according to the portion of his word which he doth grant unto the children of men, according to the heed and diligence which they give unto him.
10 And therefore, he that will harden his heart, the same receiveth the lesser portion of the word; and he that will not harden his heart, to him is given the greater portion of the word, until it is given unto him to know the mysteries of God until he know them in full.
Which is very well put. In my experience Alma's description is exactly correct. It is those that harden their hearts who don't seem to keep learning by the light of Christ and by the power of the Holy Ghost, which is always the only learning that is real gospel learning in the first place.
As to the use of the word "mysteries" in the scriptures, we find it in the first verse of the Book of Mormon. For in 1 Nephi 1:1 we read "...yea, having had a great knowledge of the goodness and the mysteries of God, therefore I make a record of my proceedings in my days".
1 Nephi 2:16 And it came to pass that I, Nephi, being exceedingly young, nevertheless being large in stature, and also having great desires to know of the mysteries of God, wherefore, I did cry unto the Lord; and behold he did visit me, and did soften my heart that I did believe all the words which had been spoken by my father; ...
1 Nephi 10:19 For he that diligently seeketh shall find; and the mysteries of God shall be unfolded unto them, by the power of the Holy Ghost, as well in these times as in times of old, and as well in times of old as in times to come; wherefore the course of the lord is one eternal round.
20 Therefore remember, O man, for all thy doings thou shalt be brought into judgment.
And so forth throughout the scriptures.