Angel, Prophet, Evangelist are really Greek words spelled in English letters. The common meanings of these words are similar, but with some important distinctions. The religious meanings are also similar and with noteworthy differences.
An angel is simply a messenger or bringer of news. The mailman and delivery person do the same job in our country today. This common meaning is used by Paul in Galatians 1:14 about himself as an angel of God in that he simply delivered Gods message to them. Likewise, this idea of people as messengers or angels of God is probably intended by the writer in Hebrews 13:2. However, in the religious sense, we usually use this word to refer to the heavenly beings sent by God to deliver His messages (and often to carry out His tasks), such as in Genesis 19:1, Zechariah 1:12-14, and Revelation 1:1. These heavenly beings never create the messages, but only deliver them.
A prophet is one who proclaims a message, much like a town-crier or an announcer. In the Bible, a prophet of God refers to a person who proclaims or delivers Gods word (sometimes, but not always, about future events). Preacher is a good synonym for prophet.
A true prophet or true preacher one who in imitating Jesus example speaks truthfully & impartially concerning Gods word and does not lie about its true meaning or his own true intentions for speaking (Luke 20:21; John 3:16-21, 4:23-24, 8:31-32, 8:44-46, 12:42-50, 17:17-21; Ephesians 4:15, 25).
A false prophet or false preacher claims he teaches concerning Gods word but his claim is false, a deception, a lie, a cheat. The apostle Paul describes such men as deceitful workers who disguise themselves as messengers of light and as servants or ministers of righteousness (2 Corinthians 11:12-15). He also describes these false teachers as sons of disobedience, lovers of themselves, lovers of money, evil men and imposters, and insubordinate, idle talkers and deceivers, who are teaching things which they ought not, for the sake of dishonest gain (Ephesians 5:6; 2 Timothy 3:1-13; Titus 1:10-11). The apostle John warns us let no one deceive you and tells us These things I have written to you concerning those who try to deceive you (1 John 3:7, 2:26; c.f. 2 John 7).
Did you notice the angel tucked inside evangelist? An evangelist is a bringer of good news. In the New Testament, the information of how to be saved is called the good news or the gospel. So, biblically speaking, an evangelist is a person who tells Gods message of how to be saved.