From Letter 246:
Frodo was sent or allowed to pass over Sea to heal him – if that could be done, before he died. He would have eventually to 'pass away': no mortal could, or can, abide for ever on earth, or within Time. So he went both to a purgatory and to a reward, for a while: a period of reflection and peace and a gaining of a truer understanding of his position in littleness and in greatness, spent still in Time amid the natural beauty of 'Arda Unmarred', the Earth unspoiled by evil.
Along with Bilbo, Gimli, and Sam (the other mortals who passed over the sea), Frodo died eventually. As Tolkien notes, he could not find peace or rest in Middle-Earth, and needed a place that might could heal him. We never really find out what he did there - it's probably sufficient to say he rested. In some ways, Aman(specifically Valinor) might could be compared to the Shire - the Shire for much of its history was unmarred by the outside world, but it couldn't compare with Valinor.
The linked question, along with this question explains in more detail, but Aman used to be accessible by normal ships, however it was forbidden for non-elves and Vala/Maia to go there. However, the Numenorians disobeyed and invaded, causing it to be removed from the "circles of the world", and only those who the Valar permitted were able to sail there, and only by a narrow path across the Sea.
The scenery of Valinor is described very limited. It was indeed a beautiful country, full of pastures and forests. The Vala made their home here, and the mythology of Tolkien is really rooted here. There once were Two Trees that gave light to the land, but the Silmarillion tells how they were poisened and how the world was changed in the following years.
In short, we don't know a whole lot about what the land looked like or what the few mortals who went there did, although we know it was a land of peace and rest, and that the mortals who did go there found relief from their hurts in their last years.