In a 1956 letter, Tolkien commented: "I found that many children have become interested, even engrossed, in The Lord of the Rings, from about 10 onwards. I think it is rather a pity, really. It was not written for them." (source, 249; in a letter to Mrs. M. Wilson, April 11, 1956).
The Hobbit, the book that comes before The Fellowship of the Ring, is a kid’s book. But The Fellowship of the Ring is not (or isn’t supposed to be). What changes between the two? We think that the difference is one of setting.
The Fellowship of the Ring starts out in much the same way that The Hobbit starts, with plenty of Hobbit-based humor. The Hobbits are hilarious, from their stodginess at Bilbo's eleventy-first birthday to their huge pleasure in drinking at the various inns throughout the Shire. The image Tolkien paints of the Shire isn't just a matter of farmland and cozy Hobbit holes. The ridiculous characters (Boffins, Chubbs, Burrows, Bolgers, Bracegirdles, etc.) are also part of the Shire's scenery, and contribute to the Shire's overall feeling of somewhat boring contentment.
Tolkien comments that the Shire represents the culture of rural England (source, 250), which explains why all of the names of the Shire have English origins – comforting counterpoint to the lovely but unfamiliar Elvish names of much of the rest of the book.
But unlike in The Hobbit (where hero Bilbo Baggins is the ultimate, representative Hobbit), we know from the first chapter of The Fellowship of the Ring that our protagonist, Frodo Baggins, is not typical of his kind. He reads Elvish and goes on long walks with Bilbo, who has been forever changed by his adventures in the first book.
Frodo certainly doesn't want to leave under the circumstances of a deadly Ring that will try to destroy his soul to return to its master, the Dark Lord Sauron. But still, the Lord of the Rings is obviously designed on a larger plan than The Hobbit, and the Shire-ness of the early chapters mostly provides contrast with the beauty and greatness of the settings of the later part of the novel.
Once Frodo leaves the coziness of the Shire, he, Merry, Pippin, and Sam immediately dive into the Old Forest. This choice of setting just after the Shire is interesting, because the Old Forest represents a completely non-human consciousness. The Hobbits of the Shire are almost too human, with their love of food, drink, jokes, and singing, and their resistance to any kind of new idea. They are extremely homey. By contrast, in the Old Forest, we have Old Man Willow, a tree filled with hatred towards all two-legged beings, and the Barrow-wights, who haunt the ancient burial mounds of the first war against the Witch-lord of Angmar. Both of these threats are not just dangerous to the Hobbits; they are also frighteningly alien.
It turns out that the landscape itself might actually be out to get them. The story continues: just walking through the land of Hollin, Aragorn warns the Hobbits: "Not all the birds are to be trusted, and there are other spies more evil than they are" (1.11.64). Similarly, Gimli talks about "Caradhras the Cruel" as though the mountain is a living thing. When the Fellowship struggles to cross the mountain passes of Caradhras thanks to a horrible, days-long blizzard, Gimli believes that the terrible storm is the work of the mountain itself. Caradhras hates the Fellowship (and all tiny, two-legged mortals) and wants them to fail. Gimli wishes, as they turn towards Moria, that Caradhras were: "'less cruel [...] There he stands smiling in the sun!'" (2.6.5).
Tolkien makes the land of Middle-earth seem like another character, one that sometimes isn't too friendly to the Fellowship. Given how much care Tolkien has lavished on his hand-drawn maps of Middle-earth, we cannot be surprised that he treats even the land itself as though it is a beloved character he has created.
Besides the Shire and the Old Forest, the other three major settings of The Fellowship of the Ring are the two Elf lands of Rivendell and Lothlórien and the now-overrun Dwarf realm of Khazad-dûm, in the Mines of Moria. These three places are much grander than the Shire, certainly. But both Rivendell and Lothlórien are nearing the end of their days. The Golden Wood of Lothlórien is fading and Khazad-dûm has been empty of Dwarves for centuries.
We get the sense, even as early as The Fellowship of the Ring, that the Elves and the Dwarves are not going to be around for long. They are starting to fade away. And even if the Hobbits can be a bit closed-minded or foolish at times, we have to admit that the Shire is a lot livelier than Rivendell, and certainly more so than Khazad-dûm. So it looks like, if the Ring quest succeeds and the side of good wins, it will be humans and Hobbits who inherit Middle-earth from these older races.