Luke mentions the circumcision of Jesus on the eighth day, in accordance with the Law. And though the place of the circumcision is not mentioned, it was likely in Bethlehem.
There is also the event of the presentation of Jesus in the Temple. The presentation of the firstborn generally took place about 40 days after birth (which Christian tradition recognizes). It seems likely that the Holy Family went to Jerusalem from Bethlehem for this. As you note, it would be unlikely that Mary, who was observing the usual period of convalescence for a woman after giving birth, would have gone all the way back to Nazareth, a round-trip journey of 140 miles.
After the Presentation we may presume that the Holy Family returned at least briefly to Bethlehem, and some time after their return the Magi made their appearance. This was followed shortly after by the flight to Egypt.
Some scholars put the visit of the Magi as late as two years after the birth of Jesus since Herod called for the death of all boys two and under. However, it is difficult to understand why the Holy Family would have stayed in Bethlehem that long, and we can reasonably assume that the Magi visited sooner than two years.
Some are troubled by the fact that Luke seems to say that they returned to Nazareth after the Presentation and that he makes no mention of the flight to Egypt, though Matthew does. But Luke’s silence about the Holy Family’s travels to Egypt does not amount to a denial of it, any more than the fact that Matthew makes no mention of the circumcision or the Presentation does not amount to a denial that they happened. Both Evangelists are merely selecting the material they deemed important.