"All they asked was that we should continue to remember the poor, the very thing I was eager to do." NIV translation
The leaders and Paul stayed focused on serving and telling people about Jesus Christ. Service to the poor is a natural thing that flows out of you because of the love of God that is in you. This is what the apostle John was talking about in his first letter to the believers (1 John 4:7).
"When Peter came to Antioch, I opposed him to his face, because he was clearly in the wrong." NIV translation
The second example takes place in Antioch which was a mixed church as it had both Jewish and Gentile believers that met together. This is a clear lesson for us about unity versus purity of the gospel. Paul did not just let Peter go on doing what he knew was wrong but instead he confronted him about his actions.
"Before certain men came from James, he used to eat with the Gentiles. But when they arrived, he began to draw back and separate himself from the Gentiles because he was afraid of those who belonged to the circumcision group." NIV translation
Now, Paul describes the problem that he had with Peter and is was simply that he was acting one way around the brothers in Antioch but another way when others showed up. In this church at Antioch, the Gentiles would eat all kinds of food that the Jews were forbidden to eat by the Mosaic Law. Peter, a Jew, knew that he was no longer under the law and that he could eat anything (because of his vision in Acts 10) and so he joined the Gentiles at the table and ate as they did. But, when the Jews from the church in Jerusalem showed up, he would not eat with the Gentiles. We, as Christians, need to be consistent in our walk and in our freedom. When someone tries to put us back under the law, we need to stand up and proclaim our freedom in Jesus Christ.
"The other Jews joined him in his hypocrisy, so that by their hypocrisy even Barnabas was led astray." NIV translation
It was bad enough that Peter was saying one thing and doing another but, by being a bad example, he had an impact on those around him. He was known to be an apostle and so was given respect and authority which caused others to do the same thing including Barnabas. That is one of the reasons why we, as Christians, should strive to be like Jesus and not any man. Jesus (the Good Shepherd) will never lead His sheep astray.