The writer of Hebrews began by proving that Jesus is superior to the angels in chapters 1-2, and not who will show that Jesus is superior to Moses. Why move from angels to Moses? For most modern readers, angels are superior to humans, so if Jesus is superior to angels, he would obviously be superior to Moses as well.
It is important to read this argument in a first century context. For Jews living in the Second Temple period, Moses was the most significant person in salvation history. There were legends developed about Moses (Assumption of Moses, for example), or Sirach 45:2, which describes Moses as equal to the “holy ones,” or even God (as the Hebrew text of Sirach can be translated, see Ellingworth, Hebrews, 194).
Sirach 45:1-2 ...and was beloved by God and people, Moses, whose memory is blessed. He made him equal in glory to the holy ones, and made him great, to the terror of his enemies.
In addition, messianic hopes in the first century sometimes focused on the coming of a prophet like Moses. Hope for a “return of Moses” as messiah was so strong that at least one messianic pretender stopped the Jordan in a re-enactment of the crossing of the Red Sea. Matthew’s gospel is designed to highlight Jesus as a new Moses. Jesus goes up on the mountain and gives the people the Law, the Sermon on the Mount!
One might have offered a counter argument to the first two chapters of Hebrews – Jesus might be superior to the angels, but the ultimate servant of God was Moses, who gave the Law. In the context of the first century, then, our author will argue that Jesus is a superior to even Moses as a servant of God. Ultimately, this will lead to the conclusion that the New covenant which Jesus enacted in his death and resurrection is superior to that of the Old Covenant made by Moses. In verse two Moses is compared to Jesus, then he is subordinated to Jesus (verse 3) and by verse 5 he is contrasted to Jesus, negatively.
This is not to say that the author of Hebrews is anti-Semitic. He is not saying that Moses is wrong, but rather than Jesus is far superior to Moses. In Hebrews, Moses and the law are good, but Jesus is better. This might have been as controversial in the first century as it is today – but to argue that Jesus and the New Covenant is superior is not to imply the Old Covenant is inferior, but dated. God is revealing himself in new ways in this present age, he has revealed himself through his Son.