Is there any indication of what Frodo and Bilbo actually did when they were dwelling in Valinor?
While it couldnt be clear what they are doing exactly there, except enjoying the presence of the Valar and the Maiar among the elves. I think that it could be answered what they are doing in a more general way.
Both Bilbo and Frodo were the Ring-bearers, so it seems that they went there as a reward and, at the same time, because it was the only site where they could live without suffering from the influence of the ring in their lives (Think about Gollum)
From the Lord of the Rings, book 6, chapter 9:
It is mentioned a few times in the text:
... for the Third Age was over, and the Days of the Rings were passed, and an end was come of the story and song of those times...
And not only Bilbo and Frodo, but even Sam is invited to go, but not now:
Then Bilbo woke up and opened his eyes. ‘Hullo, Frodo!’ he said. ‘Well, I have passed the Old Took today! So that’s settled. And now I think I am quite ready to go on another journey. Are you coming?’
‘Yes, I am coming,’ said Frodo. ‘The Ring-bearers should go together.’
‘Where are you going, Master?’ cried Sam, though at last he understood what was happening.
‘To the Havens, Sam,’ said Frodo.
‘And I can’t come.’
‘No, Sam. Not yet anyway, not further than the Havens. Though you too were a Ring-bearer, if only for a little while. Your time may come. Do not be too sad, Sam. You cannot be always torn in two. You will have to be one and whole, for many years. You have so much to enjoy and to be, and to do.’
I think that there was a text where Sam goes to Valinor. I think that Legolas went too, because he is an elf and he managed to carry Gimly with him.
Other part of the reward would be, help them to clean the corruption from the One Ring.
The most clear proof of the corruption of the One Ring is from Book 1, Chapter 1 with the confrontation between Bilbo and Gandalf:
'Everything?' said Gandalf. 'The ring as well? You agreed to that, you remember.'
'Well, er, yes, I suppose so,' stammered Bilbo.
'Where is it?'
'In an envelope, if you must know,' said Bilbo impatiently. 'There on the mantelpiece. Well, no! Here it is in my pocket!' He hesitated. 'Isn't that odd now?' he said softly to himself. 'Yet after all, why not? Why shouldn't it stay there?'
Bilbo flushed, and there was an angry light in his eyes. His kindly face grew hard. 'Why not?' he cried. 'And what business is it of yours, anyway, to know what I do with my own things? It is my own. I found it. It came to me.'
'Yes, yes,' said Gandalf. 'But there is no need to get angry.'
'If I am it is your fault,' said Bilbo. 'It is mine, I tell you. My own. My precious. Yes, my precious.'
The wizard's face remained grave and attentive, and only a flicker in his deep eyes showed that he was startled and indeed alarmed. 'It has been called that before,' he said, 'but not by you.'
'Now, now, my dear hobbit! ' said Gandalf. 'All your long life we have been friends, and you owe me something. Come! Do as you promised: give it up! '
'Well, if you want my ring yourself, say so!' cried Bilbo. 'But you won't get it. I won't give my precious away, I tell you.' His hand strayed to the hilt of his small sword.
'I don't know what has come over you, Gandalf,' he said. 'You have never been like this before. What is it all about? It is mine isn't it? I found it, and Gollum would have killed me, if I hadn't kept it. I'm not a thief, whatever he said.'
Bilbo drew his hand over his eyes. 'I am sorry,' he said. 'But I felt so queer. And yet it would be a relief in a way not to be bothered with it any more. It has been so growing on my mind lately. Sometimes I have felt it was like an eye looking at me. And I am always wanting to put it on and disappear, don't you know; or wondering if it is safe, and pulling it out to make sure. I tried locking it up, but I found I couldn't rest without it in my pocket. I don't know why. And I don't seem able to make up my mind.'
There is other example in the Book 2, chapter 1:
'And Gandalf said: "The Ring has passed on, Bilbo. It would do no good to you or to others, if you tried to meddle with it again." Odd sort of remark, just like Gandalf. But he said he was looking after you, so I let things be. I am frightfully glad to see you safe and sound.' He paused and looked at Frodo doubtfully.
'Have you got it here?' he asked in a whisper. 'I can't help feeling curious, you know, after all I've heard. I should very much like just to peep at it again.'
'Yes, I've got it,' answered Frodo, feeling a strange reluctance. 'It looks just the same as ever it did.'
'Well, I should just like to see it for a moment,' said Bilbo.
When he had dressed, Frodo found that while he slept the Ring had been hung about his neck on a new chain, light but strong. Slowly he drew it out. Bilbo put out his hand. But Frodo quickly drew back the Ring. To his distress and amazement he found that he was no longer looking at Bilbo; a shadow seemed to have fallen between them, and through it he found himself eyeing a little wrinkled creature with a hungry face and bony groping hands. He felt a desire to strike him.
Of course, you could say that the One Ring had been destroyed. But it's clear that their influence isnt going to disappear so quickly as showed for the next reason.
And that is to cure Frodo from the injury inflicted for the Nazgul, that continue disturbing Frodo two years later.
From the last chapter of book 6:
One evening Sam came into the study and found his master looking very strange. He was very pale and his eyes seemed to see things far away.
‘What’s the matter, Mr. Frodo?’ said Sam.
‘I am wounded,’ he answered, ‘wounded; it will never really heal.’
But then he got up, and the turn seemed to pass, and he was quite himself the next day. It was not until afterwards that Sam recalled that the date was October the sixth. Two years before on that day it was dark in the dell under Weathertop.
And in spite of being healed (book 2, chapter 1):
'Good!' said Gandalf. `It is mending fast. You will soon be sound again. Elrond has cured you: he has tended you for days, ever since you were brought in.'
'Days?' said Frodo.
`Well, four nights and three days, to be exact. The Elves brought you from this where you lost count. We have been terribly anxious, and Sam has hardly left your side, day or night, except to run messages. Elrond is a master of healing, but the weapons of our Enemy are deadly. To tell you the truth, I had very little hope; for I suspected that there was some fragment of the blade still in the closed wound. But it could not be found until last night. Then Elrond removed a splinter. It was deeply buried. and it was working inwards.'
Its effects are already mentioned in the book 2, chapter 4:
Frodo's spirits had risen for a while after his escape, and after food and a draught of the cordial; but now a deep uneasiness, growing to dread, crept over him again. Though he had been healed in Rivendell of the knife-stroke, that grim wound had not been without effect. His senses were sharper and more aware of things that could not be seen. One sign of change that he soon had noticed was that he could see more in the dark than any of his companions, save perhaps Gandalf. And he was in any case the bearer of the Ring: it hung upon its chain against his breast, and at whiles it seemed a heavy weight.
So, in summary, they went there to:
- As a reward for their help and
- To help them to remove the influence of the One Ring in its bearers.
And in Frodo's case:
- To heal the injury inflected by the Nazgul.