Read LOTR as if it was mythology as opposed to fantasy, which is what Tolkien was aiming for.

The reason his created 'verse is so rich is because He wrote, and re-wrote, the ancient history of Middle Earth a long time before either the Hobbit or LOTR came along. Many are written as poetic lays. In a way, those books were essentially tacked onto the end of a story arc covering thousands of years. His mythology includes the creation of the Cosmos, and the origins of all the creatures that inhabit it. The kingdoms and battles of the Third Age (when LOTR is set) are just echoes of the earlier stories. For me, that's what makes LOTR so enjoyable. Middle Earth is attractive because it has a history which is woven into the narrative. All the ruins of previous kingdoms the travelers encounter are reminders of a distant past that create a curiosity in the reader to want to learn more. One gets the feeling that the book is just scratching the surface.

Take the enigmatic Tom Bombadil, for example. Who and what the hell is he? Why is he even in the story? Firstly, Tolkien wrote a children's story about this character long before LOTR so Tolkien had him laying around and wanted to included him. But the character is also there, at that point in the story, to introduce the hobbits to an important concept in Tolkien's philosophy: that there is good in the world that cannot be corrupted. At this point in the story, the young hobbits have faced the Black Riders and the evils of the forest. They need a bit of a break and a reminder that not all is evil along their journey, "friends unlooked for.." Who is Bombadil is a topic hotly debated among fans, keeping the mystery alive. (pretty much a god/angel ("Valar") who quit his day job to live in the woods with the beautiful Goldberry.)

I agree with your suggestion that Bombadil was Tolkien finding his stride at this point in the story. The Hobbit was a book for children, a bit of a throw-off piece for his children. When the publisher wanted a sequel, he started Fellowship of the Ring, which starts out with a tone just slightly more mature than the Hobbit but gets darker and darker as the tale unfolds. You might observe that by the end of the trilogy the language gets even more Shakespearian than at the beginning.

I could ramble on more, but I'll spare you my fandom. Unless you've questions while you read, I'd be happy to answer what I can.

Meanwhile, while you read, a map is helpful, and this one is excellent:

http://lotrproject.com/map/#zoom=3&lat=-1315.5&lon=1500&layers=BTTTTT