As always, spoiler warnings...

...Frodo and Samwise rode at Aragorn's side, and Gandalf rode upon Shadowfax, and Pippin rode with the knights of Gondor; and Legolas and Gimli as ever rode together upon Arod. (Book 6, Many Partings)

Elves are strange to us, and Legolas is a strange Elf, not so much because he is an unusual Elf - there really is no evidence for that - but because of the circumstances he is placed in, traveling around with members of the other Free Peoples, a bunch of mortals with whom he shares the most significant experiences of their collective lives. It's a fantasy The Breakfast Club on the scale of Grand Romance. He is so odd he ends up best bros with a Dwarf. Following the quote at the start of the last section ('Together the Elf and the Dwarf entered Minas Tirith, and folk that saw them marveled to see such companions; for Legolas was fair of face beyond the measure of Men, and he sang an elven-song in a clear voice as he walked in the morning.'), we get:

At last they came to Prince Imrahil, and Legolas looked at him and bowed low; for he saw that there indeed was one who had elven-blood in his veins. 'Hail, lord!' he said. 'It is long since the people of Nimrodel left the woodlands of Lorien, and yet still one may see that not all sailed from Amroth's haven west over water.'   ...'I am one of the Nine Companions who set out with Mithrandir from Imaldris,' said Legolas; 'and with this Dwarf, my friend, I came with the Lord Aragorn. But now we wish to see our friends, Meriadoc and Peregrin, who are in your keeping, we are told.' (Book 5, The Last Debate)

When we first meet the Strange Elf back in Book 2, it really is all about him being a stranger to the other characters. How that has changed! Look at all the friends he mentions in these two short sentences: Mithrandir, Lord Aragorn, Meriadoc, Peregrin, and 'this Dwarf.' Perhaps no other Elf in history would be able to utter such a thing. All in just six months, less than a ripple in the ocean of his long life. (He also displays yet another superpower here - the ability to detect even minute amounts of Elvish ancestry.)

If Elrond and Galadriel and Glorfindel - not to mention Fingolfin or Luthien - are our models of Tolkien's great Elves, Legolas is clearly different: younger, less experienced, not yet a leader. But he is also presented differently, mostly in the context of non-Elf characters and their non-Elf lands. At the same time, he is still Elvish, still like Galadriel and Glorfindel, or, perhaps to put it more clearly, he shows signs that he will become like them.

Frodo looked up at the Elf standing tall above him, as he gazed into the night, seeking a mark to shoot at. His head was dark, crowned with sharp white stars that glittered in the black pools of the sky behind. But now rising and sailing up from the South the great clouds advanced, sending out dark outriders into the starry fields. A sudden dread fell on the Company.   'Elbereth Gilthoniel!' sighed Legolas as he looked up. ...Suddenly the great bow of Lorien sang. Shrill went the arrow from the elven-string. Frodo looked up. Almost above him the winged shape swerved. There was a harsh croaking scream, as it fell out of the air, vanishing down into the gloom of the eastern shore. The sky was clean again. (Book 2, The Great River)