Answer

In principle, according to Jewish law, every matter is supposed to be adjudicated in a rabbinical court. But in my opinion there is no such obligation today, for several reasons. First, regarding monetary law, the rabbinical courts do not have authority to enforce rulings, compel witnesses to appear, examine evidence, and so on. But in matters of personal status this does exist. Beyond that, in most cases the court does not rule according to Torah law, but rather according to compromise (close to the law?) and according to the law of the state. Therefore I do not see why one should specifically go to them.
By the way, there are various myths about where it is preferable to litigate, for men and for women. My understanding is that these are myths, and in many cases the rabbinical court דווקא benefits women. But that is worth clarifying with those who know the reality.