Frodo is a character fleeing from danger and into greater danger. He does take responsibility for his choices, but he is largely reactive to the greater forces acting on and around him. He loves to talk about adventure with his uncle, but his greatest achievement is his willingness to sacrifice himself. Interesting to read, yes. Interesting to play? Probably considerably less so.
Conversely, Bilbo, while initially a reluctant adventurer, really takes to it once he has decided his course. He tries to pick the troll's pockets due to a sense of professional pride. Once he has escaped the goblins, he really does consider going back under the Misty Mountains to look for his party. And, of course, he ends up having plans of his own - eventually becoming the driving force of the narrative (at least until the Battle of Five Armies). If you don't know what I am talking about here, forget the movies and read the book.
Frodo is driven by the pressures of the narrative. Bilbo takes the pressures of the narrative, and bends them to meet his personal goals. The closest that Frodo comes to this is when he chooses to accept Golum's aid - and, in the movies, this is played as though Frodo is a patsy to the wily Smeagol, whereas in the novel, he knows exactly what he is doing.
It is completely okay to play a reluctant hero. Both Frodo and Bilbo are examples of that trope, but unless you want the GM to continually drive your character's decisions, it is better to model a PC off of Bilbo than Frodo. Don't allow the needs of the Valar to move you; allow your own goals and dreams to determine what you do.
It is important to have goals.
It might not be obvious the first time one reads The Lord of the Rings - and they changed it in the movies! - but Aragorn is not a reluctant hero. He is, in fact, eager to sit on the throne of Gondor. Only when he is king can he wed Arwen, who he has loved for decades. The need to thwart Sauron is an impediment to his goals. Once he is able to finally act on them, he never loses sight of either the need to defeat Sauron or his hopes for the throne and marriage thereafter.
In game terms, a PC like Frodo is reacting to whatever the GM throws at him, whereas a Conan is actively forcing the GM to react to what he does. And an Aragorn or a Bilbo is actively trying to turn the narrative toward the direction he desires.
Have goals. It makes the game more interesting for everyone involved. And, if you have goals, the GM can use those goals as adventure hooks, making a game far more personal for your character (and, by extension, you). It doesn't even matter if your goals sometime conflict with those of your party members - that tension will add spice to the game so long as it remains in the game, and we all try to remember that everyone is there to have fun.
If you want to make the game more fun, have goals. Be prepared to add goals. Be prepared to turn the game to meet your needs. Take possession of it. Just don't do so to make it worse for the other people at the table. And be prepared to meet a particularly grisly or heroic fate if your goals demand it.
"Quest For It" is the beating heart of Dungeon Crawl Classics. Really, it is the beating heart of role-playing games. Quest for something.