1. A misinterpretation of the text.
Many often misinterpret the 1st and 15th verse of Romans 6 to mean that Paul is calling out the church in Rome for following this idea that if you sin more grace will increase, therefore sin is good. What was actually going on here is that Paul was addressing his Jewish opponents that were saying, according to Paul's gospel people would take this philosophy and thus sin more in the name of grace. This is ridiculous and if we have that mindset, we need to stop it, it is sinful.
2. You can only serve one of two masters, there is no middle ground.
We are either slaves to sin or slaves to righteousness. There is no middle ground. There has been an unbiblical concept creep into the church that there is a middle person. This person has backslid and is not living for Christ, but professed faith at some point in their life. Romans 6 says that this person is either saved or not, no middle ground.
3. What the text is not saying.
We may quickly say that than people that sin lose their salvation. No. That is not what the text is saying. Sin still exists in the life of Christians, if they say otherwise than you just caught them in sin. That is why Paul says in verse 19 that our salvation leads to sanctification. Sanctification is a lifelong process of becoming more like Christ.
4. Practical Application
The application is seen right in the text. Present ourselves as slaves to righteousness. (6:13) Live our lives for God. (6:10) Understand the parallels between Christ's life and death and our salvation presented in 5-11.
Romans 6 is a good chunk of meat, but is a refreshing break from the tension so far found in Romans. Read it. Memorize it. Live it.