Christians are "new covenant" believers. Christ gave us new laws as part of the new covenant. We don't have to obey the Law of Moses.
There is only a single place where Messiah says He gives a new commandment:
"A new commandment I give to you, that you love one another, even as I have loved you, that you also love one another. By this all men will know that you are My disciples, if you have love for one another." (John 13:34)
While love among believers is to be a sign of discipleship, it is not a new commandment. We are told, "love your neighbor as yourself" in Leviticus 19:18.
As far as the new covenant itself, God says that He will write His Law upon the hearts of those in that covenant (Jeremiah 31:31-33).
As far as the Law of Christ versus the Law of Moses, we address that claim in another article.