In Book 1 Chapter 2 , Gandalf says to Frodo that he is not warning him against leaving an address at the post office.

What did he mean by that?

'As for where I am going,' said Frodo, 'it would be difficult to give that away, for I have no clear idea myself, yet.'

'Don't be absurd!' said Gandalf. 'I am not warning you against leaving an address at the post-office! But you are leaving the Shire--and that should not be known, until you are far away. And you must go, or at least set out, either North, South, West or East--and the direction should certainly not be known.'

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There is no such line of dialogue in The Fellowship of the Ring, Book I, Chapter 2. However, we do have this exchange in Chapter 3 "Three is Company":

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Maybe you've forgotten that the Shire does maintain a postal service (the Messenger Service) with the Mayor of Michel Delving serving as the Postmaster. Gandalf is certainly aware of it, and this is what he was referencing.

And please try not to post the same question in multiple forums.

But, what I really wanted to know is that what does "leaving an address at the post office" mean in that particular situation. Taking literal meaning, what use would it serve?

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Of course, Gandalf does not think that Frodo would really leave a forwarding address. He says it himself, "Don't be absurd!" He is just reinforcing his warning to Frodo to keep his plans in the deepest secrecy.

Sorry, but what is "absurd" in:
'As for where I am going,' said Frodo, 'it would be difficult to give that away, for I have no clear idea myself, yet.'
I don't think Frodo was getting naiive...

At least in the Shire. Gandalf mentions in his letter left at Bree (Ch. 10) that Frodo's would be "Mr. Frodo Baggins, Bag End, Hobbiton in the Shire". Sam's would be more exacting - "Mr. Samwise Gamgee, No. 3 Bagshot Row, Hobbiton, The Shire".

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And Frodo at that moment didn't know where he was going, just fleeing the Shire. It's mentioned by Gandalf a couple paragraphs later that he advises Frodo to "make for Rivendell".

Having sympathetically extended his "you ought to go soon" order to "September, four months from now, is fine", Gandalf opens this next exchange with "In the meanwhile, do take care, and don’t let out any hint of where you are going!" [bold by squire]

Naturally, Frodo protests that he has no idea of where he is going, so "it would be difficult to give that away".

Gandalf's reaction, as Otaku notes, changes the subject. Frodo was not being "absurd" at all; nothing he said meant that he did not yet have a specific destination in mind, for which a forwarding address could be given. But Gandalf is crankily saying he didn't actually mean, don't say where you're going; what he meant to say was, don't say that you're going at all. He adds that, once Frodo has gone and people have noticed his absence from the Shire, it's also important that even then, Frodo should insure that the Shirefolk should have no idea in which direction he was headed -- again, useless advice from Frodo's point of view, as he pathetically reminds the wizard.

Gandalf's confusing messages show just how tense he is about this whole thing. I myself assume that he feels incredibly guilty that he has allowed the Ring to linger undetected this long in the hands of the helpless hobbits, and is projecting that anger and guilt onto Frodo.

Frodo's confession at that point, that he is completely lost and has no idea whatever of where he is supposed to go or what he is supposed to do - aside from leaving the Shire with the Ring, as he resolved at the end of the previous chapter - seems to help Gandalf to chill out, remember that hobbits are in fact quite helpless against Sauron, and actually give some workable advice: head for Rivendell.

nothing he said meant that he did not yet have a specific destination in mind, for which a forwarding address could be given

Naturally, Frodo protests that he has no idea of where he is going, so "it would be difficult to give that away".

‘I have been so taken up with the thoughts of leaving Bag End, and of saying farewell, that I have never even considered the direction,’ said Frodo. ‘For where am I to go? And by what shall I steer? What is to be my quest? Bilbo went to find a treasure, there and back again; but I go to lose one, and not return, as far as I can see.’

"‘You ought to go quietly, and you ought to go soon,’ said Gandalf. Two or three weeks had passed, and still Frodo made no sign of getting ready to go.

‘I know. But it is difficult to do both,’ he objected. ‘If I just vanish like Bilbo, the tale will be all over the Shire in no time.’

‘Of course you mustn’t vanish!’ said Gandalf. ‘That wouldn’t do at all!"

"Rivendell!’ said Frodo. ‘Very good: I will go east, and I will make for Rivendell. I will take Sam to visit the Elves; he will be delighted.’ He spoke lightly; but his heart was moved suddenly with a desire to see the house of Elrond Halfelven, and breathe the air of that deep valley where many of the Fair Folk still dwelt in peace."

'As for where I am going,' said Frodo, 'it would be difficult to give that away, for I have no clear idea myself, yet.'

'Don't be absurd!' said Gandalf. 'I am not warning you against leaving an address at the post-office! But you are leaving the Shire--and that should not be known, until you are far away. And you must go, or at least set out, either North, South, West or East--and the direction should certainly not be known.'

Realistically, Frodo only had two realistic choices of where to bring the One Ring. North was no good; what would he do, hide at the shore of Lake Nenuial? South to Saruman would have meant traveling hundreds of miles through empty lands on a poorly maintained road with a hazardous river-crossing at Tharbad. And, of course, the White Wizard had fallen to the Shadow (though neither Frodo nor even Gandalf was aware of that).

Frodo could only have taken the Ring West to Círdan in the Grey Havens, or East to Rivendell. Bringing the Ring to the Havens would actually have been a reasonable choice. Chances are good that Círdan might have taken the Ring to Rivendell himself or summoned the White Council to Mithlond. That, though, would have given us a very different story. Imagine the Ring being transported by ship to Gondor, only to be intercepted by a Corsair fleet.

could be like :

Frodo : As for where I am going, it would be difficult to give that away, for I have no clear idea myself, yet.
Gandalf : 'Don't be absurd! I am not just warning you against leaving an address at the post-office! I am warning you against even telling anyone that you are going away.

Right...?

Whether Frodo had already inwardly decided on Rivendell, I'm not sure. he has decided 'to follow Bilbo' and heading to Rivendell literally achieves this. But, as far as I know, Frodo does not know that Bilbo is there.

It's true that there are no absolutes given as to whether Frodo knows where Bilbo is, but I’ve always wondered if there might logically be some correspondence between the two. Either written or orally by messenger (wandering Elves, etc.).

They were separated almost exactly seventeen years from Sept. 22, 3001 until Frodo leaves on Sept. 23, 3018. Some simple Elf-delivered messages don’t seem beyond giving thought. Also it doesn’t seem beyond thought that Frodo in one or some of his nighttime wanderings may have come across an Elf or two that knew of Bilbo’s whereabouts. If so Frodo would know where Bilbo was.