Did the first resurrection begin with Christ? When will the 2nd begin? What will be the kingdom they will be assigned?
Indeed the first resurrection began with Christ. It was through his own power that he was able to bring such a thing about, that after death the spirit could again inhabit the body, which would be renewed to immortal life. As such, the immortal body housing the eternal spirit would never again be subjected to death. Thus, all resurrected beings live eternally and forever.
Since the resurrection of the Savior, others have been resurrected in order to perform special functions under the direction of the Savior. Such functions have included the bestowal of the keys and authorities of the holy priesthood that were held by the prophets of God in earlier dispensations of the gospel. These beings included, among others, John the Baptist, and Peter, James and John, who restored the Aaronic and Melchizedek priesthoods to Joseph Smith and his associates. Moroni was also a resurrected being, who was commissioned to give to Joseph Smith the plates upon which were inscribed the record of the Nephites, Lamanites and Jaredites, which we have as the Book of Mormon.
In addition, a general first resurrection will occur when Christ comes again to initiate his great millennial reign. In this resurrection all the good and righteous people of the earth will be brought forth as eternal resurrected beings. The evil people who are living on the earth when the Savior comes will not be resurrected at that time, but will be remanded to the custody of Satan for a thousand years. Then, at the end of the millennium they too will be resurrected. Thus every living being, without any exception, and without any regard for the kind of life they have lived, will be resurrected.
The type of resurrection and the ultimate destiny of resurrected beings can be divided into four groups. The first group comprises those who have known and loved the Savior, who have been baptized by requisite priesthood authority for the remission of their sins, and who have been true and faithful to all the covenants encompassed in the gospel of Jesus Christ throughout their mortal lives. They will be resurrected with celestial bodies to a celestial glory, and will live in a celestial kingdom with God and Christ forever more.
The second group who will come forth in the first resurrection are all the good people of the earth, but who chose not to follow the Savior as members of his kingdom on the earth, and thus made no covenants with the Savior of obedience to the principles of his gospel. They will receive a terrestrial resurrection, inherit a terrestrial glory, and will live on a terrestrial world, which is the type of a world that this earth was before the fall of Adam and Eve.
The third group to be resurrected will be those wicked people who followed Satan rather then the Savior and were under Satan’s power and dominance during the thousand years of the Lord’s millennial reign on the earth. They will be resurrected with a telestial glory and with telestial bodies, and will inhabit a telestial world. This telestial world with be of the same nature that our earth is at this time, it now being a telestial body. However, the glory of this telestial world will be so great as to surpass all understanding.
There is yet a fourth group to be resurrected. They are those who, having known the Savior by the ministrations of the Holy Spirit and having made covenants of obedience to his gospel, have then turned away and have become the enemies of righteousness. These will also come forth in the second resurrection at the end of the millennium. But they will not inherit a kingdom of glory, but will be cast into outer darkness, where they will live in agony with Satan for the rest of never-ending eternity. These are called the sons of perdition.
You may read all about it in the Section 76 of the Doctrine and Covenants.