Answer: There are more than twenty interactions humans have experienced with angels listed both in the Gospel accounts and in the rest of the New Testament. The below list of angelic appearances is listed in chronological order.
The first New Testament interaction with an angel happens to Zacharias in Jerusalem's temple. He is told his wife Elizabeth will bear a son whose name will be John (John the Baptist). John 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).
Amazingly, Joseph receives at least three separate visits from angels. He received one regarding marrying Mary and two (somewhat later) that revolve around the protection of Jesus from Herod (Matthew 1:18 - 20, 2:12 - 13, 19 - 21).
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 righteous spirits also praise God for the unique miracle of Christ's birth to a virgin (Luke 2:9 - 15).
The New Testament also records a group of angels ministering to Jesus after His temptation by Satan the devil (Matthew 4:11).
From time to time an angel would stir the water in the pool of Bethesda. The first person to enter the pool after the water was stirred would be healed of their diseases (John 5:1 - 4).
God sent a spiritual messenger to Jesus to strengthen him before his suffering and death. The Bible states, just after Christ urged the disciples to pray so that they did not fall into temptation, "Then an angel appeared to Him from heaven, strengthening Him" (Luke 22:43).
An angel appears twice near Jesus' tomb declaring, to Mary, Mary Magdalene and others, that the Lord has already risen from the dead (Matthew 28:1 - 2, 5 - 6, Mark 16:5 - 6). 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 messenger also states that Paul's prayer that all those aboard the ship are saved is granted (Acts 27:23 - 24).
One of the greatest New Testament interactions with an angel occurs when one is sent to the Apostle John. He journeys to the apostle, 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).
The last time an interaction with angels is recorded in the New Testament is when one informs John that all the prophecies he saw are faithful and will come to pass. John is also warned not to worship angelic spirits but only God (Revelation 22:6 - 11).