"Missionary" is generally used to mean someone who has made converting others to his religion his life's work. The usual connotation is that this is his job: he is supported by a church or missionary organization or some form of contributions from others. If he does have a regular job to support himself, he views this as secondary: his job is a means to support his mission work, and not something he does for itself. (Missionaries who support themselves are often called "bivocational missionaries".
"Evangelist" is very similar in meaning to "missionary". In Christian circles, I think the word "missionary" is used for people who go to a foreign country or another culture, while "evangelist" is used for those who try to spread the faith at home. Like speaking as someone from the U.S., a person who toured the U.S. preaching Christianity would be called an "evangelist", while someone who went to Kazakhstan to spread Christianity would be called a "missionary".
"Proselytize" is a verb meaning to try to convert others to your religion. The dictionary gives "proselytizer" as the noun form, but I think that's very rarely used. "Proselytize" tends to have negative connotations. If you resent someone else trying to convince you to change your religion, you might say "he's trying to proselytize me". But a religious group would be unlikely to say, "Hey, let's go out and try to proselytize a bunch of people".
In conservative Christian circles, it's common to refer to people trying to convert others to Christianity as "witnessing", and to a person who engages in such efforts as "a witness". As in, "Bob is trying to be a witness at his job", meaning, Bob is trying to talk about Christianity and convert others at his job. I've never heard this term used for other religions: I don't think anyone talks about a "Muslim witness" or a "Buddhist witness" in this sense.
There's also the general word "preacher". A "preacher" might be trying to convert people. But he also might be trying to teach people who already share his faith more about that faith, or encourage them to practice it more faithfully, etc.
Anyone who talks a lot about his faith might be called an "outspoken X", that is, an "outspoken Christian", "outspoken Muslim", etc.