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[LotR] Why didn't Frodo just walk west? What would have happened if he had?

I've just started rereading *Fellowship of the Ring*, and there's a thing in Chapter 3 that I'd completely forgotten about - right before Gandalf goes off, he and Frodo make plans for how the hobbit is going to move the Ring out of the Shire. 

Frodo, sort of sentimentally, decides to follow Bilbo's route, and Sam's just excited to have a chance to meet Elves. The decision is kind of lightly taken - at one point Gandalf asks (paraphrasing), "Which way do you think you'll go? Any way will do as long as it's not here."

So fast forward a few months (into the next chapter or two), Gandalf never shows up, Frodo, Pippin and Sam find themselves dodging a Black Rider or two in the less settled areas of the Shire, and then are lucky enough to come across a party of High Elves - who they get along great with. Sam's overjoyed, Frodo speaks their language, they give him a little news about Bilbo, it's all pretty grand. 

They're heading west to the Grey Havens. All the High Elves do. 

What if Frodo did, too? The elves weren’t sure where Bilbo was, but knew he was hanging with other elves. Frodo has no reason to go south (where the Enemy is), but also no real reason to head east toward the mountains – not yet, anyway. Just a kind of notion. Gandalf’s not there to give advice, and his generally cheery outlook about heading toward Rivendell might well be misplaced given that he’s missed his appointment and put the hobbits in jeopardy. 

The one place where they know lots of elves are going is not too far outside the Shire, and pretty ideal if you wanted to get as far from the Enemy as possible – they’ve got ships!

Why not go west *to start with*? Frodo’s even a Ring-Bearer, so entitled to free passage (although he wouldn’t know that yet.)

How would things unfold then?
gpt
He's definitely not entitled to free passage to the Havens at that point in time.

Frodo's (and Bilbo's, possibly Sam's) acceptance to the Havens and on the boats west was granted to them *after* the destruction of the ring, as a reward to their service to Middle-Earth and as a solace for their ring-depression, which they could never shake as long as they lived in Middle-Earth.

Before the war of the ring, there wasn't any sort of invitation, and as you mentioned, Frodo wouldn't have known about it anyways.

But more important is the issue of the Ring, not just the Bearer.  The Elves were very, very hesitant to let the Ring stay in their safe havens.  In both Rivendell and Lothlorien, the mood was pretty much "You're here so we can keep you safe for a few weeks, then you and that Ring need to get the hell out."  Rivendell, Lorien, and the Havens were the last bastions of real Elvish power, kept safe from Sauron's corruption through the power of the Three Rings.  Bringing the One to any of them was a danger, and was only done for safety and to aid the Bearer and his Fellowship.  There was definitely no way to leave the Ring at one of them, or to bear it across the sea.