Depending upon who you ask, the series begins with The Lion, the Witch, and the Wardrobe OR it begins with The Magician's Nephew. I prefer to begin with The Magician's Nephew as it's technically the prequel (although written second to last) and sets up the creation of Narnia (the lamppost!). It follows Digory and Polly as they are forcefully transported to Narnia by Digory's not-so-nice aka totally evil uncle, Before they reach Narnia, however, the reach other lands in other realms and on one of these they meet Jadis who is Evil. (The capitalization is definitely warranted and she's worth mentioning as you'll see later in the series.) This is also the book that introduces the reader to Aslan, the lion.
Next is the one I think most people are familiar with: The Lion, the Witch, and the Wardrobe. In this book, we are introduced to the Pevensie children: Peter, Susan, Edmund, and Lucy. We're also meeting Digory again except now he's much older and a Professor. We travel to Narnia which is now fully formed (and through a rather unique portal which I don't want to spoil for the uninitiated) and a battle of good vs evil ensues against the White Witch (hello Jadis!). The children discover that they are worth sterner stuff than they had imagined and Aslan rewards them with something a little snazzier than a plaque. This should definitely be categorized as coming-of-age.
I bet you think that Prince Caspian is the next book in the series. Well, you're wrong! It's The Horse and His Boy. This book follows two Narnians (the horses) on a journey back to their homeland with outsiders (two children) who are running away from their lives. This story is quite different from the rest of the series as it takes place during the Golden Age of Narnia (it is difficult not to be spoiler-y isn't it?). It focuses on Shasta, Bree, Aravis, and Hwin. If nothing else, C.S. Lewis was a master at names.
Following The Horse and His Boy is Prince Caspian. By this time, the Pevensie children have returned to our world and a short time has passed. They often think of Narnia but for Peter and Susan it is less and less like a real place. Then BOOM they are suddenly thrust back into Narnia as a cry for help is sounded by Caspian. 1000 years have passed in Narnia and Aslan is missing. The people are in dire need of help and it's up to the Pevensies and their new friend, Caspian, to save everyone.
The Voyage of the Dawn Treader is a favorite of mine because it introduces a character which goes through such a transformation that to meet him at the start is to hardly recognize him by the finish. This is Eustace Scrubb. Eustace and his two cousins, Edmund and Lucy Pevensie, find themselves in Narnia and on a boat in the ocean. There's a surprise awaiting them aboard the ship in the form of an old friend (hello spoiler, my old friend) and it's discovered that he and his crew are on an epic quest. It's very nearly a pirate's tale, ya'll. The ending still manages to bring me to tears.
We are again on an adventure with Eustace in The Silver Chair but this time he's accompanied by his new friend, Jill Pole. They are called to Narnia by Aslan to help Caspian, now an old man, find his son. They are aided by Puddleglum who is a Marsh-Wiggle and if you don't grit your teeth he might drive you crazy.
Here we are at the end with The Last Battle which is just as the name suggests. There is a final battle of good vs evil. Jill and Eustace return to help Narnia which is under seige by Shift, an ape, who tricks a donkey into impersonating Aslan (I just read that back and trust me it makes sense if you read the book). By the end of the book you see the true meaning behind everything that has gone before and Aslan is revealed as his true self.