Jesus is setting "his face to go to Jerusalem" because the time's coming "for him to be taken up" (9:51).
Did you catch that? This verse signals a new sub-section of Luke's story, which focuses on Jesus's journey from Galilee in the north, through Samaria, Judea, and finally to Jerusalem in the south. The whole journey motif will be around until his arrival in Jerusalem in 19:28.
Jesus sends his emissaries before him to prepare his own arrival.
They enter a Samaritan village, which is not hospitable to Jesus because the inhabitants don't like his destination: Jerusalem.
Take a time out to understand that at this time, Samaritans were religiously and ethnically distinct from the Jews in Galilee and Judea. They recognized the authority of the first five books of the Hebrew Bible alone and considered Mt. Gerizim in Samaria—not Jerusalem—to be their sacred center. Their origin is a matter of debate, but severe tensions between Samaritans and Jews had long been in the making by the time of Jesus.
Back to the story: James and John ask Jesus if they should curse the village to destruction by fire.
Jesus reprimands them. Apparently it's a bad idea, and so they move on to the next village.
That was random.
While journeying on the road, someone volunteers to follow Jesus wherever he goes.
Jesus is an honest guy, so he clarifies for the guy what he's about to get himself into.
Even foxes and birds have places to hang their hats. But "the Son of Man has nowhere to lay his head" (9:58). The implication is that following Jesus will require this guy to be homeless.
One guy's on board, and now Jesus orders another guy to follow him. But guy #2 asks for some time to see to his father's funeral.
Evidently, followers have to skip even that kind of thing: "Let the dead bury their own dead" (9:60). Have fun teasing that one out.
A third person commits to follow Jesus as soon as he's said farewell to his family.
For Jesus, this is a sign of lack of commitment. After all, "no one who puts a hand to the plow and looks back is fit for the kingdom of God" (9:62).
Conclusion? Jesus is on the move now and so are his followers. They'll have to embrace homelessness and can't even leave a note with the fam.