Question: "What does the Bible say about lawsuits / suing?"
Answer: The apostle Paul instructed the Corinthian believers to not go to court against one another (1 Corinthians 6:1-8). For Christians not to forgive each other and reconcile their own differences is to demonstrate spiritual defeat. Why would someone want to become a Christian if Christians have just as many problems and are just as incapable of solving them? However, there are some instances when a lawsuit might be the proper course of action. If the biblical pattern for reconciliation has been followed (Matthew 18:15-17) and the offending party is still in the wrong, in some instances a lawsuit might be justified. This should only be done after much prayer for wisdom (James 1:5) and consultation with spiritual leadership.
The whole context of 1 Corinthians 6:1-6 deals with disputes in the church, but Paul does reference the court system when he speaks of judgments concerning things pertaining to this life. Paul means that the court system exists for matters of this life that are outside the church. Church problems should not be taken to the court system, but should be judged within the church.
Acts chapters 21—22 talk about Paul being arrested and wrongfully accused of a crime he did not commit. The Romans arrested him and "the commander brought Paul inside and ordered him lashed with whips to make him confess his crime. He wanted to find out why the crowd had become so furious. As they tied Paul down to lash him, Paul said to the officer standing there, "Is it legal for you to whip a Roman citizen who hasn't even been tried?" Paul used the Roman law and his citizenship to protect himself. There is nothing wrong with using the court system as long as it is done with a right motive and a pure heart.
Paul further declares, "Actually, then, it is already a defeat for you, that you have lawsuits with one another. Why not rather be wronged? Why not rather be defrauded?" (1 Corinthians 6:7). The thing Paul is concerned with here is the testimony of the believer. It would be far better for us to be taken advantage of, or even abused, than it would be for us to push a person even further away from Christ by taking him/her to court. Which is more important—a legal battle or the battle for a person's eternal soul?
In summary, should Christians take each other to court over church matters? Absolutely not! Should Christians take each other to court over civil matters? If it can in any way be avoided, no. Should Christians take non-Christians to court over civil matters? Again, if it can be avoided, no. However, in some instances, such as the protection of our own rights (as in the example of the apostle Paul), it may be appropriate to pursue a legal solution.