This phrase is repeated throughout the book, generally in regards to Mr. Tulliver. "Going to law" is essentially a sort of old-fashioned way of saying "getting involved in legal proceedings," or a lawsuit. Mr. Tulliver is of course oddly fixated on the law. We start hearing about his hatred of lawyers from the first chapter, where he notes that they were created by the devil. And it is his involvement with a lawsuit that leads to a dramatic reversal of the Tulliver family fortunes.
So what is so significant about the law, besides its role as a plot device? Well, Mr. and Mrs. Tulliver are characterized in terms of their relationship to the law. Both view the law and legal proceedings in old-fashioned terms. Mrs. Tulliver seems to have some sort of superstitious aversion to the law, not understanding it beyond a vague notion that it causes problems. Mr. Tulliver also seems rather mystified by the law. He wants Tom to get a good education so that he can be on par with smarty-pants lawyers, after all. Mr. Tulliver also views the law in very superstitious, old-fashioned terms, given his views on how the devil created lawyers. Mr. Tulliver’s thoughts on the law shed light on his character:
Mr. Tulliver was on the whole a man of safe traditional opinions; but on one or two points he had trusted to his unassisted intellect and had arrived at several questionable conclusions, among the rest, that rats, weevils, and lawyers were created by Old Harry [the devil]. (1.3.2)
The law basically stands in for modern life in general, which the old-fashioned Tullivers find really confusing. The ways that characters view the law represents their non-modern or else modern understandings of the world. What does it mean to have a modern world view? Here’s an explanation, using some metaphors:
It is clear that the irascible miller was a man to interpret any chance shot that grazed him as an attempt on his own life, and was liable to entanglements in this puzzling world which [...] required the hypothesis of a very active diabolical agency to explain them. It is still possible to believe that [Wakem] was not more guilty towards him, than an ingenious machine which performs its work." (3.7.10) (To see this quote in full with commentary check out the "Society and Class Quotes.")
The law here is basically an impersonal machine, part of a modern world of factories and banks and lawsuits. Characters like the Tullivers are sort of stumbling around in this world blindly. Mr. Tulliver still sees the world in terms of the devil, "Old Harry," while Mr. Wakem sees the world in terms of profits and clients and impersonal business transactions. Wakem is a machine doing his job. This is probably why Mr. Wakem assumed Mr. Tulliver was drunk right before Tulliver attacked him; having some sort of blood-feud just isn’t Wakem’s style, since it's not modern. Wakem takes revenge by buying people’s property, not beating them up. Overall, the law represents modern society and helps us to better understand character’s places in this "puzzling" world of lawsuits and legal procedure.