Alma 13:1-5/Priesthood and rest
The discussion of the priesthood in verses 1-20 may be read as a clarification of the promise to enter into the rest of God which Alma extended along with the invitation to repent in Alma 12:37. Alma 13:6 describes the task of preaching as teaching others so that "they also might enter into his rest." The implication of the word "also" here is that those preaching have already entered into his rest. In this case, ordination itself--through the preparatory redemption (see comments on the main Alma 13:1-5 page for more on preparatory redemption)--is the process of entering into the rest of God. This is confirmed by verses 10-12. There Alma describes those who "were ordained and became high priests of God," and, "being called after this holy order," they "were sanctified," "their garments were washed white," and they were thereby "made pure and entered into the rest of the Lord their God." These passages seem quite clearly to mean that ordination to the high priesthood is the way to enter into the rest of the Lord.
Another way of reading this passage is that the high priesthood was the means by which others could enter the rest of the Lord. The epistle to the Hebrews is another notable sermon that includes a discussion on entering the rest of the Lord. Interestingly it also includes a discussion on the priesthood of Melchizedek, just as Alma's sermon does. The references in Hebrews seem to support the idea that the Melchizedek priesthood enabled others to enter God's rest, but focuses more on the direct role of the Savior rather than of ordained high priests.