If you are interested it isn’t absolutely necessary to go back to the beginning. The first episode of the third season (listed as Episode 27 on Crunchyroll) does an adequate recap of the premise, and the individual episodes stand by themselves well enough that they can be enjoyed without knowing everything that happened before.
Takashi Natsume (Hiroshi Kamiya) is an orphan boy who often feels isolated because of his ability to see spirits. His grandmother Reiko (Sanae Kobayashi) had the same problem when she was his age. She dealt with it by challenging various youkai to a test of strength, and when she defeated them she would force them to write their names in her “Book of Friends.”
When Takashi inherits the book he becomes the target of powerful youkai who are determined to possess it, since its owner can control the youkai whose names are written in it. Fortunately he gains the help of a powerful cat-like youkai named Madara (Kazuhiko Inoue). He promises Madara, whom he calls “Nyanko-sensei”, that he can inherit the book if he protects Takashi while he is alive.
Takashi seems to have better social skills than his grandmother and many of the stories involve him making friends of both the human and supernatural variety. The series is largely about the importance of friendship and family. (Of course I could probably make the case that every anime ever made is ultimately about that.)
For fans of the first two seasons the third offers some treats. It finally reveals the nature of the contests in which Reiko defeated the youkai–which actually puts her story in a somewhat different light.