What Happens to Legolas in Return of the King?

In The Return of the King, Legolas and his best friend Gimli go with Aragorn to aid Gondor. There they fight in the great battle to save Minas Tirith from Sauron's armies. Afterwards they follow Aragorn to the Black Gates of Mordor to fight another (nearly suicidal) battle. That keeps the bulk of Sauron's forces in the north, away from Frodo's path to Mt. Doom.

Trivia note: In that exact same spot, Legolas' grandfather died leading a brazen charge against the Black Gate in the Last Alliance. That's when Legolas' father, one of the few survivors of that charge, became King of Mirkwood.

After the final victory, Legolas and friends stay in Minas Tirith through Aragorn's coronation and wedding. Then they go Rohan for Théoden's funeral and the wedding of Éowyn and Faramir. At Isengard, the Fellowship parts at last: the Hobbits head back to the Shire, Aragorn returns to his kingdom, and Legolas and Gimli journey home together to Mirkwood and the Lonely Mountain.

The Appendices of The Return of the King tell us two things about Legolas' fate:

First, he takes a group of Elves to settle in Ithilien, the forest across the river from Minas Tirith where Frodo met Faramir.

Second, after Aragorn dies of old age, Legolas sails to the Undying Lands, where presumably he still lives.

"When King Elessar gave up his life Legolas followed at last the desire of his heart and sailed over Sea. We have heard that Legolas took Gimli Glóin's son because of their great friendship, greater than any that has been between Elf and Dwarf. If this is true, then it is strange indeed: that a Dwarf should be willing to leave Middl-earth for any love, or that the Eldar should receive him, or that the Lords of the West [the Valar] should permit it. But it is said that Gimli went also out of desire to see again the beauty of Galadriel; and it may be that she, being mighty among the Eldar, obtained this grace for him."(Appendix A 'One of the last notes in the Red Book', The Return of the King)
(Gimli would've died of old age sooner or later. Even in the Undying Lands, mortals don't become immortal, they just have all hurts and sorrows healed.)

There is one other passage about Legolas' fate in the "Epilogue" of The Lord of the Rings, a chapter that Tolkien dropped at the urging of his friends. Here, Sam answers some questions his children have written for him to help him finish writing Bilbo and Frodo's book:

Q: Legolas. Did he go back to the King? Will he stay there?
A: Yes, he did. He came back south with Gimli, and he brought many of his people from Greenwood the Great (so they call it now). They say it was a wonderful sight to see companies of Dwarves and Elves journeying together. The Elves have made the City, the land where Prince Faramir lives, more beautiful than ever.Yes, Legolas will stay there, at any rate as long as Gimli does; but I think he will go to the Sea one day."(Epilogue, The End of the Third Age)