ANSWER: Your question revolves around the preaching, in pairs, that the twelve disciples carried out roughly two years after Jesus' ministry began in 26 A.D. (Mark 6:7 - 13). They were sent on what we would call a supervised training mission so that they could gain the experience and skills needed for preaching the gospel after the Lord's death.
Jesus initiated another training exercise in preaching roughly one year after the first one mentioned above. For this training, however, he gathered seventy of his disciples and sent them out in pairs to several locations. The goal was to give them practical experience in evangelism (preaching) and to prepare the people for a visit by him (Luke 10:1 - 20).
The use of pairs for preaching and serving continued to be used after Christ's death. For example, after Paul and Barnabas were ordained apostles (Acts 13:1 - 3), they evangelized several places during what is called Paul's first missionary journey. A few years later, Paul and Silas preach together during Paul's second missionary journey while the pair of Barnabas and Mark revisits Christians on the island of Cyprus.
Why did Jesus send out his disciples out in pairs? Although Scripture does not directly reveal his reasons, it was likely done so that a credible witness of his entire ministry, teachings and miracles could be offered. He intentionally choose the disciples with this goal in mind (John 3:11, 15:27, Acts 1:22).
The testimony of one person is not considered acceptable by God (see 1Corinthians 14:29, 2Corinthians 13:1). In the Old Testament, the testimony of a single person was not considered sufficient to convict someone of murder (Numbers 35:30, Deuteronomy 17:6). In fact, two or three witness were required in order to establish that a person committed a sin worthy of punishment (Deuteronomy 19:15).
Another reason He sent the disciples out in pairs was so that there would be a credible witness against those who rejected their preaching.
Christ told his disciples to reject those who refused to hear what they had to say and, as a warning against them, to shake the dust off their feet as they left (Mark 6:11). This symbolized that even the earth (the dust) was a witness against those who refused to hear the preaching of God's truth (see Deuteronomy 30:19).
Should pairs still be used today for preaching and serving others? A team of two people counselling others, visiting those interested in the Bible and so on offers further credibility and authority to what is said. Utilizing two people also offers the opportunity to insure the needs of those seeking help are best served. Lastly, a pair of people who visits someone in their home reduces the risk of being falsely sued or blackmailed.