ANSWER: There are at least twenty distinct interactions between angels and humans in the Gospel accounts and the New Testament as a whole. The below list of angelic appearances is listed in chronological order.
An angel appears to Zacharias, in Jerusalem's temple, announcing his wife Elizabeth will bear a son whose name will be John. He will have the Holy Spirit from his mother's womb and live like a Nazarite (Luke 1:11 - 20, 26 - 38).
Gabriel (who is one of a class of angels called Archangels) is sent to a virgin named Mary to inform her that she will miraculously conceive of the Savior whom will be named Jesus (Luke 1:26 - 38).
An angel communicates to Joseph, in a dream, that he should not fear remaining married to Mary (he should not divorce her, Matthew 1:20).
An angel announces to Bethlehem's shepherds that Jesus has been born. They are also told where to find the newborn King and Savior of mankind. Multiple angels also praise God for the unique miracle of Christ's birth to a virgin (Luke 2:9 - 15).
An angel communicates to Joseph, in a dream, warning him to flee with his family to Egypt. He is also told to stay in the country until the threat from Herod, to destroy Jesus, has passed (Matthew 2:13).
An angel communicates to Joseph, in a dream, that Herod the Great, who sought to murder his child, is now dead. Joseph is told it is now safe for him and his family to travel back into Israel (Matthew 2:19).
A group of angels minister to Jesus after His temptation by Satan the devil (Matthew 4:11).
An angel is sent by God the Father to strengthen Christ while he prays and waits for his arrest in the Garden of Gethsemane (Luke 22:43).
An angel appears, at Jesus' tomb, to Jesus' mother, Mary Magdalene and others informing them that the Lord has risen from the dead. He also tells them to let the other disciples know of his resurrection and that he will meet them in Galilee (Matthew 28:2 - 7).
Two angels, who look like men, appear to the eleven disciples on the Mount of Olives right after Jesus' ascension into heaven. They inform them that Christ will return to the earth in the same way he left (Acts 1:10 - 11).
Jewish religious leaders, in Jerusalem, arrest the twelve apostles and put them in prison. God sends the angel of the Lord to free them from prison. After the disciples are freed they are encouraged to boldly continue to preach the gospel (Acts 5:17 - 21).
An angelic being appears to Philip the Evangelist and commands him to travel to Gaza. During his travel he runs into an Ethiopian eunuch, explains the gospel to him, and ultimately baptizes him (Acts 8:26 - 38).
An angelic being appears to a Roman Centurion named Cornelius, in a vision, informing him to seek out the Apostle Peter. Cornelius and his family are baptized, becoming the first non-Jewish converts to Christianity (Acts 10:3 - 7, 30 - 32).
After Peter is thrown into prison by Herod Agrippa I God sends an angel to free him and lead him to safety (Acts 12:1 - 10).
An angel appears to Paul, in a dream, while he is sailing as a prisoner to Rome. He is told that he will not die during the journey but rather will appear before Caesar. The angelic messenger also states that Paul's prayer that all those aboard the ship are saved is granted (Acts 27:23 - 24).
An angel is sent to the Apostle John, who has been exiled to the island of Patmos, to reveal prophecies to him that will ultimately become the book of Revelation (Revelation 1:1).
The Apostle John, in a vision, takes a prophetic little book from the hand of an angel. The spirit being tells him, "Take it and eat it; and it shall make your belly bitter, but in your mouth it shall be sweet as honey" (Revelation 10:8 - 9, HBFV).
An angel tells John to take a reed and measure God's temple (Revelation 11:1 - 2).
An angel reveals to John the true meaning of a woman, riding a scarlet beast, who has on her forehead "MYSTERY, BABYLON THE GREAT, THE MOTHER OF THE HARLOTS AND OF THE ABOMINATIONS OF THE EARTH" (Revelation 17).
An angel informs John that all the prophecies he saw are faithful and will occurcome to pass. John is also warned not to worship angelic spirits but only God (Revelation 22:6 - 11).