Go rather to the lost sheep of Israel.
1. Verse 10:6 serves as an important exegetical focal point for three reasons: First, it presents the difficulty of understanding who are "the lost sheep of Israel", and how they are lost. Second, this verse interacts (as Luz points out, p. 73) with both 15:9 and the mission command in 28:19. Third, and related to the first two, is the focus on Israel over and against the Gentiles.
2. Both Luz (p. 73) and Carter (p. 234) understand the verse, with regards to the first difficulty, as inclusive of all of Israel, not simply "the sinners, outcasts, and marginalized in Israel" (Luz, p. 73). They both also relate 10:6 to the others mentioned above and while Carter believes that the focus on Israel alone is temporary, and will come to include the Gentiles as well, Luz is of the opinion that the move to the Gentiles in 28:19 is the result of a "change of direction" made by Jesus, because Israel has rejected him.
3. From preliminary readings, I tend to agree with Carter on all three points, although I would want a more nuanced explanation of the inclusivity of his understanding of the first problem of the verse. It seems to me that to just say "the lost sheep of Israel" means all of Israel does not appreciate the differentiation present at that time between "the tax collectors and sinners" and the Pharisees (and everyone in between, for that matter...). What did it mean to be a faithful Jew in Jesus' time? Surely not all of Israel can be divided into these two classes. All this raises questions of what it means to be a part of Israel, and what it means to be lost. These questions are complicated when read in light of Ephesians, particularly the first two chapters, for example. Finally, to say with Luz that the mission to the Gentiles is a rejection of those who have rejected Jesus is a grievous misreading.