For the story of the three witnesses we will turn to Joseph's record in History of the Church:
"In the course of the work of translation, we ascertained that three special witnesses1 were to be provided by the Lord, to whom He would grant that they should see the plates from which this work (the Book of Mormon) should be translated; and that these witnesses should bear record of the same, as will be found recorded, Book of Mormon, page 581 [Book of Ether, chapter 5, verses 2, 3 and 4, p. 487, edition 1920], also page 86 [II Nephi, chapter 11, verse 3, p. 73, edition 1920].2 Almost immediately after we had made this discovery, [p.53] it occurred to Oliver Cowdery, David Whitmer and the aforementioned Martin Harris (who had come to inquire after our progress in the work) that they would have me inquire of the Lord to know if they might not obtain of him the privilege to be these three special witnesses; and finally they became so very solicitous, and urged me so much to inquire that at length I complied; and through the Urim and Thummim, I obtained of the Lord for them the following:
Revelation to Oliver Cowdery, David Whitmer, and Martin Harris, at Fayette, Seneca County, New York, June, 1829, given previous to their viewing the plates containing the Book of Mormon."3
Joseph next records the 17th section of the D&C, which begins as follows:
"Behold, I say unto you, that you must rely upon my word, which if you do with full purpose of heart, you shall have a view of the plates, and also the breastplate, the sword of Laban, the Urim and Thummim, which were given to the brother of Jared upon the mount, when he talked with the Lord face to face, and the miraculous directors which were given to Lehi while in the wilderness, on the borders of the Red Sea."
The Lord continues to explain that they shall obtain this blessing by faith, they shall testify of the things they shall see, and that if they are faithful, they shall be lifted up at the last day (see DC 17). Joseph continues his history:
"Not many days after the above commandment was given, we four, viz., Martin Harris, David Whitmer, Oliver Cowdery and myself, agreed to retire into the woods, and try to obtain, by fervent and humble prayer, the fulfillment of the promises given in the above revelation-that they should have a view of the plates. We accordingly made choice of a piece of woods convenient to Mr. Whitmer's house, to which we retired, and having knelt down, we began to pray in much faith to Almighty God to bestow upon us a realization of these promises.
"According to previous arrangement, I commenced vocal prayer to our Heavenly Father, and was followed by each of the others in succession. We did not at the first trial, however, obtain any answer or manifestation of divine favor in our behalf. We again observed the same order of prayer, each calling on and praying fervently to God in rotation, but with the same result as before.
"Upon this, our second failure, Martin Harris proposed that he should withdraw himself from us, believing, as he expressed himself, that his presence was the cause of our not obtaining what we wished for. He accordingly withdrew from us, and we knelt down again, and had not been many minutes engaged in prayer, when presently we beheld a light above us in the air, of exceeding brightness; and behold, an angel stood before us. In his hands he held the plates which we had been praying for these to have a view of. He turned over the leaves one by one, so that we could see them, and discern the engravings thereon distinctly. He then addressed himself to David Whitmer, and said, 'David, blessed is the Lord, and he that keeps His commandments;' when, immediately afterwards, we heard a voice from out of the bright light above us, saying, 'These plates have been revealed by the power of God, and they have been translated by the power of God. The translation of them which you have seen is correct, and I command you to bear record of what you now see and hear.'
"I now left David and Oliver, and went in pursuit of Martin Harris, whom I found at a considerable distance, fervently engaged in prayer. He soon told me, however, that he had not yet prevailed with the Lord, and earnestly requested me to join him in prayer, that he also might realize the same blessings which we had just received. We accordingly joined in prayer, and ultimately obtained our desires, for before we had yet finished, the same vision was opened to our view, at least it was again opened to me, and I once more beheld and heard the same things; whilst at the same moment, Martin Harris cried out, apparently in an ecstasy of joy, "'Tis enough; 'tis enough; mine eyes have beheld; mine eyes have beheld;" and jumping up, he shouted, 'Hosanna,' blessing God, and otherwise rejoiced exceedingly.4
"Having thus, through the mercy of God, obtained these glorious manifestations, it now remained for these three individuals to fulfill the commandment which they had received, viz., to bear record of these things; in order to accomplish which, they drew up and subscribed5 the following document:" (He then records the Testimony of the Three Witnesses, see History of the Church, vol. 1, pp. 52-56)
The three witnesses also see the sword of Laban, the Liahona, the Urim and Thummim, and the breastplate as promised in DC 17:1
Although Joseph does not describe this event in the History of the Church, David Whitmer is quoted as follows:
"We not only saw the plates of the Book of Mormon but also the brass plates, the plates of the Book of Ether, the plates containing the records of the wickedness and secret combinations of the people of the world down to the time of their being engraved, and many other plates ... there appeared as it were, a table with many records or plates upon it, besides the plates of the Book of Mormon, also the Sword of Laban, the Directors i.e., the ball which Lehi had-and the Interpreters [Urim and Thummim]. I saw them just as plain as I see this bed (striking the bed beside him with his hand), and I heard the voice of the Lord, as distinctly as I ever heard anything in my life declaring that the records of the plates of the Book of Mormon were translated by the gift and power of God." (1878 interview between Orson Pratt and David Whitmer, recorded in Book of Mormon Compendium, pp. 55-56)
Joseph's joy that others had seen the plates and been charged to testify of them
After the heavenly vision, the four men returned to the Whitmer farm. The prophet's mother was there at the time and records:
"When they returned to the house it was between three and four o'clock p.m. Mrs. Whitmer, Mr. Smith and myself, were sitting in a bedroom at the time. On coming in, Joseph threw himself down beside me, and exclaimed, 'Father, mother, you do not know how happy I am: the Lord has now caused the plates to be shown to three more besides myself. They have seen an angel, who has testified to them, and they will have to bear witness to the truth of what I have said, for now they know for themselves, that I do not go about to deceive the people, and I feel as if I was relieved of a burden which was almost too heavy for me to bear, and it rejoices my soul, that I am not any longer to be entirely alone in the world.' Upon this, Martin Harris came in: he seemed almost overcome with joy, and testified boldly to what he had both seen and heard. And so did David and Oliver, adding that no tongue could express the joy of their hearts, and the greatness of the things which they had both seen and heard." (History of Joseph Smith by his Mother, p. 153)