The new testament does not have much good to say in general about going to court. Consider the following scriptures:
25 Come to terms quickly with your accuser while you are going with him to court, lest your accuser hand you over to the judge, and the judge to the guard, and you be put in prison. 26 Truly, I say to you, you will never get out until you have paid the last penny. Matthew 5:25-26.
1 When one of you has a grievance against another, does he dare go to law before the unrighteous instead of the saints? 2 Or do you not know that the saints will judge the world? And if the world is to be judged by you, are you incompetent to try trivial cases? 3 Do you not know that we are to judge angels? How much more, then, matters pertaining to this life! 4 So if you have such cases, why do you lay them before those who have no standing in the church? 5 I say this to your shame. Can it be that there is no one among you wise enough to settle a dispute between the brothers, 6 but brother goes to law against brother, and that before unbelievers? 7 To have lawsuits at all with one another is already a defeat for you. Why not rather suffer wrong? Why not rather be defrauded? 8 But you yourselves wrong and defraud—even your own brothers! 1 Corinthians 6:1-8.
6 But you have dishonored the poor man. Are not the rich the ones who oppress you, and the ones who drag you into court? James 2:6.
On the positive side, the Roman legal system and Paul’s rights under it was used of God to save Paul from the jews. I’m sure that there are some other positive examples.
Nevertheless, the overall emphasis of these scriptures is to be wary of going to court. In the Matthew 5 verses, the point is to not be presumptious in taking someone to court. Instead, be humble and seek to work out your differences with people rather than taking them to court. In the 1 Corinthians 6 passage, Paul is exhorting the Christians to turn away from suing one another in secular courts. Why not rather be defrauded! And in the James verse, we see how courts can often be the tool of the rich to the detriment of the poor. This continues to be so today because of the difficulty of going to law without an attorney and attorneys can only really be afforded by a small percentage of the American population.
Is this to say that the civil courts should never be used. No, we still need to render to Ceaser the things that are Ceaser’s, and for family law, the state has a legitimate interest in our families and the break up of the family. But the civil courts need to be used much more as a last resort then is prevalent today with Christians turning first to God’s word and the church to help them resolve disputes, including family disputes.