The person doing so is called a convert.
Conversion requires internalization of the new belief system.
Christianity and Islam are two religions that encourage preaching their faith in order to convert non-believers. In the year 1000, the viking age parliament of Iceland decided that the entire country should convert to Christianity, and that sacrifice to the old gods, while still allowed, should no longer be made in the open. Similar mass conversions in other Scandinavian countries were not as democratic.