According to the record in Luke 2:21-39, Jesus was brought to the Temple in Jerusalem 8 days after his birth, afterwards the family returned to Nazareth.
In Matthew 2:14-23 it is recorded that the family fled to Egypt after the birth of Jesus and was there until the death of Herod. When returning from there they passed by Judea and then went on to Nazareth.
These remarks about the apparent contradiction between these two records about events after the birth of Jesus attest to a rather superficial reading which results in assumptions, and these assumptions are what causes the apparent contradiction. The two biblical records are otherwise very clear and contain no contradiction whatsoever.
Both records contain information by which it is possible to set them chronologically in relation to the birth of Christ. The result of such consideration is that the record in Luke 2:21-38 preceeds the record in Matthew 2:14-23. Luke 3:29 then is sort of a summarizing statement which emphasizes that they fulfilled all that was required by the Law of the LORD and afterwards returned to their hometown of Nazareth.
Luke 2:21 records the circumcision and naming of Jesus which happened 8 days after the birth according to the ordinance given in the Law. The record actually doesn't say that they went to Jerusalem and into the Temple for that event. The next event, recorded in Luke 2:22-24, the presentation of Jesus happened in Jerusalem in the Temple, and it happened "when the days of her [Mary's] purification were accomplished according to the law of Moses". This was 40 days after the birth in the case of a boy child (cp Lev 12:2-4). Joseph and Mary brought Jesus 40 days after the birth to Jerusalem in order to fulfill what was required by the Mosaic Law in regards to the birth of firstborn son. On that day they met Simeon and Anna in the Temple, which event is recorded in Luke 2:25-38. They had come to Jerusalem from Bethlehem and obviously returned back to Bethlehem afterwards. At this time and most likely for some time after this the family lived in Bethlehem, but no longer in the inn where Jesus had been born but rather in a different house.
This is confirmed in the record about the visit of the wise men in Matthew 2:1-12. They came some time after the birth of Jesus to Jerusalem and travelled from there to Bethlehem where they found the child in "a house". The word for "child" used here in the Greek text, paidion, does not refer to a newborn baby (as does the word brephos used in Luke 2:12); this also indicates that the wise men did not come to the house immediately after the shepherds. The visit of the wise men was only a relatively short time before the death of King Herod, and from the mention that Herod sought to kill all male children of an age "two years and under" it can be concluded that Jesus was no longer a newborn baby. After the visit of the wise men the family departed into Egypt in obedience to the instruction given by the angel where they remained until after the death of Herod.
When they returned from Egypt the plan may at first have been to return again to Bethlehem. But because of the situation with Archelaus they did not travel into Judea but rather continued on to Nazareth in Galilee in order to live there.
This return to Nazareth is also spoken of in the statement in Luke 2:39. This verse does not follow immediately from a chronological standpoint after the events recorded in the verses just before this. This verse is rather a sort of summarizing statement about the truth that they fulfilled all that we required in the Law and in the prophets which included the flight to Egypt and the return from there (cp. Mat 2:15). After this all had been fulfilled, the family returned to Nazareth by which again another prophecy of the prophets was fulfilled (cp. Mat 2:23).