The first resurrection to which Alma refers is the resurrection which took place at the time of Christ. It included the resurrection of all the righteous who died before the resurrection of Christ. Joseph Fielding Smith gives us the names of some of the righteous who were resurrected at this time: Adam, Eve, Abel, Seth, Noah, Shem, Abraham, Isaac, Jacob, Moses, Isaiah, Ezekiel, Daniel, Elias, Malachi, and the prophets who dwelt among the Nephites (DC 138:38-49). Alma himself was surely part of this first resurrection.
The resurrection of the righteous at the time of Christ happened both in the Old and the New World.
’And the graves were opened; and many bodies of the saints which slept arose,
And came out of the graves after his resurrection, and went into the holy city, and appeared unto many.’ (Matt 27:52-3)
’Verily I say unto you, I commanded my servant Samuel, the Lamanite, that he should testify unto this people, that at the day that the Father should glorify his name in me that there were many saints who should arise from the dead, and should appear unto many, and should minister unto them. And he said unto them: Was it not so?
And his disciples answered him and said: Yea, Lord, Samuel did prophesy according to thy words, and they were all fulfilled.
And Jesus said unto them: How be it that ye have not written this thing, that many saints did arise and appear unto many and did minister unto them?
And it came to pass that Nephi remembered that this thing had not been written.’ (3 Nephi 23:9-12)
By definition, the resurrection which is associated with the Second Coming is also referred to as the first resurrection.
Bruce R. McConkie
"To those who lived before the resurrection of Christ, the day of his coming forth from the dead was known as the first resurrection. Abinadi and Alma, for instance, so considered it. (Mosiah 15:21-25; Alma 40.) To those who have lived since that day, the first resurrection is yet future and will take place at the time of the Second Coming. (D. & C. 88:96-102.) We have no knowledge that the resurrection is going on now or that any persons have been resurrected since the day in which Christ came forth excepting Peter, James, and Moroni, all of whom had special labors to perform in this day which necessitated tangible resurrected bodies.
“Though all men are assured of a resurrection, all will not be resurrected at the same time, and there will be varying degrees of glory for immortal persons. All will come forth from the grave, ‘But every man in his own order’ (1 Cor. 15:23), as Paul expresses it.” (Mormon Doctrine, p. 639-40)
Joseph Fielding Smith
"While there was a general resurrection of the righteous at the time Christ arose from the dead, it is customary for us to speak of the resurrection of the righteous at the Second Coming of Christ as the first resurrection. It is the first to us, for we have little thought or concern over that which is past. The Lord has promised that at the time of his Second Advent the graves will be opened, and the just shall come forth to reign with him on the earth for a thousand years…
"RESURRECTION OF CELESTIAL BODIES. In modern revelation given to the Church, the Lord has made known more in relation to this glorious event. There shall be at least two classes which shall have the privilege of the resurrection at this time: First, those who ‘shall dwell in the presence of God and his Christ forever and ever’; and second, honorable men, those who belong to the terrestrial kingdom as well as those of the celestial kingdom (DC 76:62,75).
"At the time of the coming of Christ, ’They who have slept in their graves shall come forth, for their graves shall be opened; and they also shall be caught up to meet him in the midst of the pillar of heaven-They are Christ’s, the first fruits, they who shall descend with him first, and they who are on the earth and in their graves, who are first caught up to meet him; and all this by the voice of the sounding of the trump of the angel of God.’ (DC 88:97-8)…
"RESURRECTION OF TERRESTRIAL BODIES. Following this great event, and after the Lord and the righteous who are caught up to meet him have descended upon the earth, there will come to pass another resurrection. This may be considered as a part of the first, although it comes later. In this resurrection will come forth those of the terrestrial order, who were not worthy to be caught up to meet him, but who are worthy to come forth to enjoy the millennial reign…(DC 88:99)
"RESURRECTION OF TELESTIAL BODIES. All liars, and sorcerers, and adulterers and all who love and make a lie, shall not receive the resurrection at this time, but for a thousand years shall be thrust down into hell where they shall suffer the wrath of God until they pay the price of their sinning, if it is possible, by the things which they shall suffer.
"These are the ’spirits of men who are to be judged, and are found under condemnation; And these are the rest of the dead; and they live not again until the thousand years are ended, neither again, until the end of the earth.’ (DC 88:100-1)
“These are the hosts of the telestial world who are condemned to ‘suffer the wrath of God on earth’; and who are ’cast down to hell and suffer the wrath of Almighty God, until the fulness of times, when Christ shall have subdued all enemies under his feet, and shall have perfected his work.’ (DC 76:104-6)” (Doctrines of Salvation, vol. 2, pp. 295-7)