For a three-act plot analysis, put on your screenwriter’s hat. Moviemakers know the formula well: at the end of Act One, the main character is drawn in completely to a conflict. During Act Two, she is farthest away from her goals. At the end of Act Three, the story is resolved.
Act I
Frodo Baggins is a prosperous Hobbit in the peaceful, pleasant land of the Shire (though his neighbors think he's kind of odd). He inherits a golden ring from his uncle, Bilbo Baggins, the famous Hobbit adventurer. How could inheriting a gold ring be a bad thing? When it is the Ruling Ring of the Dark Lord Sauron, who will stop at nothing to get it back. Frodo has to leave the Shire fast, accompanied by his gardener, Sam, and his cousins, Merry and Pippin. The four Hobbits do their best to cross the wilderness with the help of their new friend and companion, Aragorn, but they still run into trouble: Frodo is stabbed by the Lord of Sauron's worst servants, the Ringwraiths, and almost dies before making it to the Elf stronghold of Rivendell with the Ring. In Act I of the novel, Frodo is demonstrating two of his traits: (1) he doesn't necessarily have an instinct for adventure, and he makes tons of mistakes. But (2), he is tough and determined when it matters most, more so than a lot of more heroic-looking folk might be.
Act II
In general, the second act is the moment in the book when we are as far from the plot's resolution as we are going to get. In The Fellowship of the Ring, that moment comes during the Council of Elrond. There, we hear about all of the bad omens that are rocking Middle-earth: for example, the Dwarf king Dáin has received messengers from Sauron looking for the Ring, and Boromir and his brother, Faramir, have had prophetic dreams about "Isildur's Bane" (the Ring) in Rivendell. Elrond and Gandalf are absolutely certain that the side of good cannot use the Ring without bringing ruin on themselves. But then, what do they do with the Ring? This section of the novel ramps up the suspense for the whole rest of the series. It lays out that the good side has no choice but to destroy the Ring, while making it clear that getting to Mordor to toss the thing into the fires of Mount Doom will be extremely tough.
Act III
So now we know that (a) Frodo is not a big adventurer, so he is an unlikely candidate for Ring-bearer; and (b) taking the Ring to Mordor is going to be an impossibly difficult job for anyone to undertake. How the heck is Frodo supposed to accomplish this task? Because The Fellowship of the Ring is only the first book of three, we just have to get to the end of the beginning, to the point where Frodo is truly ready to shoulder the responsibility of being Ring-bearer. He spends the later chapters, from the departure from Rivendell to Amon Hen, building up the confidence and conviction to decide where to go with the Ring. But it isn't until Boromir tries to snatch the Ring from Frodo that Frodo fully realizes how dangerous the Ring is, and how carefully he has to keep it even from the well-meaning members of his Fellowship. After all, characters like Boromir and Aragorn both have business in Gondor, with the war against Sauron. They cannot be as focused on the Ring as Frodo has to be. Once Frodo and Sam leave the Fellowship behind, that is when the Ring quest into Mordor truly begins, with their single-minded travel into the land of the Enemy.