'In this mythical 'prehistory' immortality, strictly longevity co-extensive with the life of Arda, was part of the given nature of Elves; beyond the End nothing was revealed.'
'They are therefore 'immortal'. Not 'eternally', but to endure with and within the created world, while its story lasts. When 'killed' by the injury or destruction of their incarnate form, they do not escape from time, but remain in the world, either discarnate, or being re-born. This becomes a great burden as the ages lengthen, especially in a world in which there is malice and destruction.'
JRRT, selections from Letters
As for the Undying Lands -- the land does not confer immortality (strictly longevity co-extensive with the life of Arda) on Mortals. And the Elves are 'immortal' already in Middle-earth, though they will ultimately fade in the body if they do not go to the Undying Lands. It can get more detailed than this, but this is a general description anyway.
By 're-born' Tolkien means reincarnated (that is, at least he ultimately would 'mean' reincarnated).
Of course if you need to get to the bookstore or the library, you might need a car too
Seriously, though. The Silmarillion is a must-read for true Tolkien fans. It's the ultimate legendarium, and it certainly covers the longest period of history in the world he created. Get it!
Or... hint it for a gift on a special day?