I would say that Judaism is first an ethnic group, as it is more the culture you are born into or choose to practice that makes you Jewish than whether you actually believe in the Torah and Yahweh (otherwise there would be no such thing as a ‘jewish agnostic’).
And it also has elements, but not all the necessary elements, of being considered a race or a religion. ‘Race’ implies a common root, and indeed, all Jewish communities or at least most of them originated from the ancient Israelis, the thing is, since you can convert to Judaism, even though most Jews are born and DO have an Israeli bloodline, since you can convert, not all of them do. So for that reason only you can’t call Jewish people a racial group.
Judaism is not a religion either. Well, a part of it is a religion, but the Jewish religion is just the core part of the Jewish community. Due to the presence of Jewish atheists, agnostics and converts to other faiths who are still considered Jewish because of their background, you can’t say Judaism is a religion in its entirety.
So I would say Judaism is an ethnic group, and a Jew is a member of this ethnic group, either (usually) by birth and thus, of Israeli ancestry dating back to the people in the Bible, or by choice. In fact, i might even say there are several Jewish ethnic groups, because Ashkenaz, Sephardic, etc are all quite different.