After Jesus "set His face" to go to Jerusalem (remember that 9:51 is the turning point in Luke's Gospel account), we see things intensifying towards that end. You'll notice in this chapter that conversations and events relating to the religious leaders of the day are getting more confrontational, until a climactic statement at the end of the chapter. You'll also notice an increase in the crowds following Jesus (which is no doubt a big factor in the angst of the religious leaders).
A few notes on the text:
11:1-13 Praying
When Jesus' disciples asked Him how to pray, Jesus gives them a model. This is quite different from a formula or mantra, where you'd need to say the exact words to make it "work". Prayer is what's been called "answering speech" to God; it's not a formula. Even though almost all Christians have learned the Lord's Prayer in "King James" English (that is, from the translation of the King James Version of the Bible), the reason we all say it the same is for one of ease, to be able to pray it out loud together. It can most certainly be also prayed in more contemporary English (or any other language!). The point is that we have a model prayer from Jesus Himself, and so we can always be assured that God will answer that prayer with a resounding yes!
After giving the model for prayer, Jesus goes on to teach more about prayer, pointing out that Christians can indeed be bold in prayer before God, knowing that God will hear them, and will grant them all the requests that are in line with His will.
11:14-26, 29-36 Continued Exorcisms and Increasing Opposition
Notice again that there are many receiving these healings with wonder. But there is now growing opposition (11:15-16). It doesn't make much sense that Jesus would be a demon who cast our demons, and Jesus calls them out not only on that, but on their desire for some greater sign. What greater sign could they be looking for? It seems that nothing would really satisfy them, since these requests wren't coming in faith but in unbelief. And unbelief will find a way to not believe, no matter the evidence.
For those seeking a sign (11:16, 11:29-30), Jesus will not accept their testing, but will give them the only sign that matters: the sign foreshadowed by Jonah. And in fact it will be a greater sign, though not one that these challengers are seeking. This recalling of Jonah—and then telling them that He Himself is greater than Jonah—points to His own three-day ordeal. But it also points to the message and preaching of Jonah, which is the same as John the Baptist and Jesus Himself: repent and be saved. The people of Ninevah did and were, but those who are opposing Jesus—those who refuse to repent and receive that salvation (because they refuse to recognize their need for it)—will not be.
11:27-28 Blessings in Truth
A most interesting interjection here. Amid opposition, Luke shares a woman's blessing of Jesus. We recognize that this woman is speaking of Mary, Jesus' mother. Remember that part of the Magnificat, the song of Mary (1:48), where Mary praised God because all generations would call her blessed? That's coming true even here!
Jesus acknowledges that yes, Mary is blessed, but maybe not in the way we might think. It might come across as harsh to our ears, but Jesus isn't denying that Mary is blessed. It's just the reason that she's blessed isn't because she's His mother. True, from an earthly perspective it is a blessing to be a mother. But Jesus is teaching that earthly families do have a proper perspective (recall also 8:21); earthly families—while important and called together as families by God—are secondary to the family of God. And so Mary more properly blessed because 11:28 applies to her! She is one who heard the Word of God and kept it. Remember her response to the Annunciation? “Behold, I am the servant of the Lord; let it be to me according to your word” (1:38)!
11:37-53 Trying to Defend Hypocrisy
One theme we haven't so far focused on is one of table fellowship. In the Middle Eastern context, to eat with someone was to show deep fellowship with them. To host someone for a meal was not just to provide food and drink, but also to offer protection and to signify a deep relationship. It would seem natural to Pharisees that Jesus would dine with them; He was a Rabbi (Jewish teacher), after all. But His eating with others would also cause scandal (it becomes the catalyst for the famous parable of the father and sons in Luke 15).
So for Jesus to call out the hypocrisy of the Pharisees, and then more specifically the lawyers (not what we would understand as solicitors or barristers in a court of law, but rather experts in the Jewish Torah), is one thing. To call out their hypocrisy while being hosted at a formal meal would be all the more insulting. Understanding this context—as well as the actual content of Jesus' pronouncement of woes—helps us understand why these Jewish leaders were beginning to oppose Him so strongly. He comes into their midst, passes over the ceremonial laws of ritual washing, and then insults their application of Torah.
The thing is, it's not an insult; it's truth-telling. They were seeing Torah not as a joyful response to the salvation that Yahweh has brought (e.g. as we hear in Psalm 119), but as a burden of rules. Even worse, they made others follow these rules that they themselves did not! So Jesus, as He so often did, was simply correctly teaching and applying Torah, and it's with heavy irony that the Pharisees and experts in Torah miss that fact.
And instead of listening to this blessed Rabbi, they dig in their heels against Him. They "had it out" for Jesus, and they began to actively try to trap and discredit Him. This will continue to intensify as Jesus' journey towards Jerusalem continues.