Remember that even though we largely deal with books of the Bible according to chapters and verses, these aren't part of the original text. So in many cases, our starting and stopping points by chapters aren't the most logical places in terms of context. This is another place where that's very true, as the context of the preceding chapters flows into this one. In particular, the first part of chapter 13 is highly dependent on the end of chapter 12, where Jesus has been teaching the crowds in an extended discourse. Though that discourse ends here at 13:1 and the specific topic shifts, the fact that Luke tells us that "There were some present at that very time…" teaches us that this is a continuing conversation, not a new account, as the chapter begins.
13:1-9
We dealt with this more in a recent sermon, so we won't include extensive notes here, but the general theme here is considering one's own need for repentance and forgiveness, rather than focusing on the sins of others and measuring God's favour to them in terms of earthly outcomes.
13:10-20 Another Sabbath Controversy
Jesus healing people is something we've seen much of in Luke's account, since it goes hand-in-hand with Him teaching as He brings the Kingdom of God to the people. Though we dare not take this healing lightly—it is a miracle, and what a relief to this woman after 18 years of suffering!—we also need to note that one emphasis here is on the timing of the miracle, rather than the miracle itself. It's quite amazing that the ruler of the synagogue where this healing happened didn't really think the healing was a big deal, only that he thought it shouldn't have happened on the Sabbath.
The Sabbath was a prescribed day of rest for the people of Israel—prescribed by God Himself!—but like with so many things given by GOd, Israel's rulers had lost sight of its true purpose. Jesus, as He so often does, here again corrects the misunderstanding and mis-application of Torah. As He says elsewhere (e.g. Luke 6:5, in the first recorded Sabbath controversy in Luke), Jesus is also Lord of the Sabbath, and the Sabbath was given as a gift to people, not for people to serve it (Mark 2:27).
In addition, this was a pre-figuring of the giving way of the Sabbath to a focus on the day of eternal healing, the day of resurrection. The Old Covenant Sabbath day was what we know as Firday evening through Saturday evening (days were counted from dusk-dusk in the Hebrew way of counting them, not midnight-midnight like we do), and that gives way to "the Lord's Day", the day of the resurrection of Jesus, what we know as Sunday. This day is the "eighth day" of creation, the day of new creation, the first day of the week (e.g. Luke 24:1).
Jesus tells parables in support of this pre-figuring, showing that these seemingly small, "hidden" things will one day become fully formed. For now, the Kingdom of God seems so small, like a mustard seed or a bit of yeast. But one day it will be fully realized.
13:22-35 Salvation Through the Narrow Door
This passage can cause people some distress, because it can seem like salvation isn't a sure thing. Some may read this as "we tried our hardest and did all the right things, but we still weren't saved". So it causes concern and doubt, where really there needn't be that.
Here Jesus is not saying, "you might not be saved, even though you did all the right things". He's saying, "you were given ample opportunity to enter—that is, to confess Jesus as Lord and Messiah and to turn to Him by faith for your salvation—but you refused. And there will be a day when it will be too late for that to happen".
This connects us back to the parable of the fig tree in the vineyard (13:9). God's patience for people to come to repentance is exceedingly generous. But there will be a day where that comes to an end. In general, that is the end of our earthly lives, unless He returns prior to that for any given person (we'll see that more in Luke 16 with the parable of the rich man and Lazarus).
This refusal to receive Him as Messiah is also the reason for His lament over Jerusalem. Again, we dealt with these verses more in a recent sermon. God takes no joy in sinners not being saved; He desires that all come to a knowledge of Jesus as the Way, the Truth, and the Life (1 Tim. 2:4, John 14:6). And so Jesus weeps over those who reject Him and His saving work for them.