Frodo Baggins is quite literally called to the Ring quest: Gandalf finds him in his comfortable home in the Shire and warns him that he must leave right away because servants of the worst evil in Middle-earth are looking for him personally, as the (extremely) unexpected owner of the most powerful magic ring in the world.
Frodo's true journey in The Fellowship of the Ring takes place when he chooses to leave the Shire without Gandalf's careful assistance: he chooses to save the Shire from the attentions of Sauron by removing himself from it as fast as he can. But even though he has great intentions, he manages to get into quite a bit of trouble, both in the Old Forest and on Weathertop. These struggles force Frodo to confront his own inexperience as an adventurer, leaving his future as Ring-bearer somewhat in suspense.
Frodo reaches Rivendell at death's door, thanks to a shoulder wound from the cursed Morgul-knife of the Lord of the Ringwraiths. But Elrond manages to cure him (mostly). So now Frodo has delivered the Ring to Rivendell, as promised. He has drawn the Ringwraiths away from the vulnerable Shire. Now what? Surely he can't be expected to keep carrying the Ring all the way to its destruction in Mordor? But the thing is, the only person who can destroy the Ring is one who has no ambition to use it from the start. The greater the wisdom or power of the bearer, the stronger the Ring's temptations would be. Frodo is a humble person from a small place: he gives the Ring less to work with. So even though the first part of the quest is finished, Frodo's arrival in Rivendell signals the start of the second, much longer and more dire part of his journey.
If there's one thing Frodo's disastrous journey through the Old Forest has taught us, it's that he needs some instruction before he is ready to lead the Ring quest on his own. But Frodo's biggest ordeal in the second half of the book – except maybe the death of his beloved mentor, Gandalf – is the face-off with Boromir. In fact, both the death of Gandalf and the disaster with Boromir serve similar plot purposes: to teach Frodo the importance of making decisions and standing on his own two feet. When Boromir tries to take the Ring, Frodo has to confront the power of the Ring over everyone else in the Fellowship. His ultimate responsibility is to destroy the Ring, but it isn't until Boromir attacks that Frodo realizes he can't hand that responsibility off to anyone else.
There is still a lot of quest left in the Lord of the Rings series after the end of The Fellowship of the Ring, so Frodo doesn't achieve his ultimate goal at the end of this book. But he does achieve the goal of deciding, at last, to take the Ring quest into his own hands. Gandalf and Aragorn both give great advice, but this quest is ultimately Frodo's. He is the one who fate has chosen to accept this awful burden. His plan to go into Mordor with only Sam by his side at the end of The Fellowship of the Ring indicates Frodo's final acceptance of his extremely dangerous destiny.