The second journey began on a rather unfortunate circumstance: "Barnabas wanted to take John, also called Mark, with them, but Paul did not think it wise to take him, because he had deserted them in Pamphylia and had not continued with them in the word. They had such a sharp disagreement that they parted company. Barnabas took Mark and sailed for Cyprus, but Paul chose Silas and left." (Acts 15:37-40)
The reason for the quarrel was that during Paul's First Missionary Journey, the young Mark had left Paul and Barnabas and returned home early in the journey. Although Paul and Barnabas apparently never again traveled together, there was no lasting animosity between them - Paul later spoke highly of Barnabas. Paul also made up with Mark, who later was with him during Paul's imprisonment in Rome (Colossians 4:10, 2 Timothy 4:11).
Paul's second missionary journey began about 49 AD, and like the first journey, it was no "10-day excursion." He would not return for about 3 years, until 52.
The previous journey began by sailing to Cyprus, but this time he set out overland through Syria and Cilicia, to revisit the churches he had earlier established in Asia, including those at Derbe, and then at Lystra where Timothy joined him (Acts 16:1-5). From there they continued northward through Phrygia and Galatia (Acts 16:6). Paul remained in Galatia for some time due to an unspecified illness (Galatians 4:13-14).
From Galatia, Paul then intended to travel northeastward through Bithynia, a region on the shore of the Black Sea, however "they tried to enter Bithynia, but The Spirit of Jesus would not allow them to. So they passed by Mysia and went down to Troas." (Acts 16:7-8). Troas was on the shore of the The Aegean Sea - Jesus Christ Himself was directing Paul westward from Asia, where Paul had intended to remain, toward Greece. Paul was taking the Gospel across to Europe.
At Troas, "During the night Paul had a vision of a man of Macedonia standing and begging him, "Come over to Macedonia and help us." (Acts 16:9). Paul realized that the vision was a message from The Lord, so the very next day he sailed across the Dardanelles, also called the Hellespont, which separated him from Europe. It was a big step for Christianity that The Spirit of Jesus Christ Himself was guiding.
In Macedonia, churches were established through Philippi (Acts 16:11-39), Thessalonica (Acts 17:1-9), and Berea (Acts 17:10-15). The journey was not without its problems - Paul and Silas were often opposed by those who refused to believe, and were once thrown in jail for casting the spirit out of a Fortune Teller (Acts 16:16-40). It seems that riots and mobs were always close at hand.
To escape the deadly threats made against him, "the brothers immediately sent Paul to the coast, but Silas and Timothy stayed at Berea. The men who accompanied Paul brought him to Athens and then left with instructions for Silas and Timothy to join him as soon as possible." (Acts 17:14-15). It was then, while waiting for them, that Paul made his now-famous visit to Athens (see Paul In Athens).
From Athens Paul crossed over to Corinth, the seat of the Roman government of Achaia. He remained there for a year and a half, spreading The Word to Jews and Greeks with much success. Also at Corinth, he wrote his two epistles to the church of Thessalonica, his earliest apostolic letters - today known as First and Second Thessalonians (see By The Book).
After numerous other spiritually profitable visits throughout the region, Paul then began making his homeward journey because he wanted to observe Pentecost at Jerusalem. He was accompanied by Aquila and Priscilla as far as Ephesus. From there he made his way by sea, through Rhodes and Cyprus to Caesarea. From there he went up to Jerusalem.
See also On The Road To Damascus and Paul's Third Missionary Journey.
Fact Finder: Paul's 3 missionary journeys were a very important part of Bible History (see also Bible Places). The account of the 3 missionary journeys, and his later being taken to Rome as a prisoner (and suffering his famous shipwreck along the way) are highly recommended reading.
The first journey: Acts 13:1 to 15:35
The second journey: Acts 15:36 to 18:22
The third journey: Acts 18:23 to 21:17
The voyage to Rome: Acts chapters 27 and 28