The short version is: Part 2 is a significantly better read than Part 1, but you really have to read Part 1 to understand Part 2.
Part 2 starts with all those formalities that Part 1 lacked. The dedications in particular. And here Cervantes introduces us to an incident that evolves into a bit of an obsession throughout the work. After Part 1 became a success, someone else wrote a sequel without Cervantes's permission. He deemed it quite inferior, obviously, both in writing style and character development. The irony, of course, is that we're still reading Cervantes's Don Quixote 400 years later, and we would never even remember this counterfeit if he didn't harp on it so much.
In this world of Part 2, we encounter our Knight of the Rueful Countenance (later to be known as the Knight of the Lions) still in bed recovering from his second sally, just a month or so before. And somehow in this time, half of Europe has read the true history of Don Quixote (the one by Cide Hamete Benegeli, of course, the historian Cervantes claims to be translating through both parts) and is completely enamored with the protagonist and his squire. They discuss the short-comings of the book--specifically the crazy novel-within-a-novel part I complained about earlier and a plotting mistake--and it's funny to hear that the complaints of readers 400 years ago are so similar to today's.
Don Quixote and Sancho Panza decide to proceed on a third sally, and they sneak out of the house in order to do so. Since I promised to keep this brief, I won't recount their adventures here. The important difference between this and the first and second sallies is that they actually win sometimes. The reader cheers for them because not every adventure ends in a spectacular beating or a sheep slaughter. Not all (most) of their wins are objectively fair successes, but they think they're winning, and you can't help but want it for them. A lot of people play a lot of tricks on them throughout, but many of the tricks are harmless except that they increase the delusion of Don Quixote and Sancho.
We love them for their delusions. When Sancho occasionally contemplates walking away, the reader wishes him to stay. As mean as some of the tricks are, we are the trick-players. We want them to believe, maybe because we want to live in a world where their beliefs are relevant.
And that's the magic of Part 2. Despite all the cultural differences between early 17th century Spain and 21st century America, we can still relate to the emotions behind the characters and their stories. Maybe this is what earns Don Quixote the title of "the first modern novel," I'm not sure. But it feels modern.