Canto 1 is the introduction, by Sūta. Here he meets Śaunaka and the Naimiṣāraṇya Sages, starts answering their questions, decides to narrate the Bhāgavatam to them, and answers their initial question about the three main people behind the Bhāgavatam (Vyāsa, Śuka and Parīkṣit).
Canto 2 is another introduction, this time by Śuka. Here Śuka meets Parīkṣit, starts answering his questions, and decides to narrate the Bhāgavatam, gives the summary of it by explaining its original seed form (four-verses).
Canto 3 begins the Bhāgavatam itself. This canto has three stories: Vidura’s pilgrimage, Varāha’s adventures, and Kapila’s teachings, in that order. The main story is Kapila’s teachings, in my opinion.
Vidura meets Uddhava in Vṛndāvana, finds out that Krishna has left the world, and asks to be taught the knowledge Krishna gave Uddhava. But Uddhava requests him to learn it from Maitreya. The conversation between Vidura and Maitreya will span all of Canto Three and Four. Vidura’s questions to Maitreya allow Maitreya to talk about Varāha and Kapila.
Vidura asks how the universe began and evolved into the shape its in today. Maitreya explains this, and comes to the part where the neglected, uninhabited earth disappears to the nether regions and has to be reclaimed and restored by Varāha. He mentions that Varāha does this on a few occasions, and once he had to kill a terrible demon to do it. He explains who this demon is, so we hear about Jaya and Vijaya being curesed by the four Kumar.
After this first Varaha incarnation, the universe began to be successfully populated. We hear the story of the Progenitors who helped make the possible, Kardama and Devahūti. Their child is Kapila, who enlightened his mother, and exactly how he did so is explained in detail.