Magic as it relates to the Elves is a somewhat cloudy subject. Galadriel comments to Sam,
This [the Mirror of Galadriel] is what your folk would call magic, I believe; though I do not understand clearly what they mean; and they seem also to use the same word of the deceits of the Enemy. But this, if you will, is the magic of Galadriel.
(Lord of the Rings, Book III, Chapter 7, "The Mirror of Galadriel")
Elves use a couple of different procedures that Tolkien has different names for:
Magic should be reserved for the operations of the Magician. Art is the human process that produces by the way (it is not its only or ultimate object) Secondary Belief. Art of the same sort, if more skilled and effortless, the elves can also use, or so the reports seem to show; but the more potent and specially elvish craft I will, for lack of a less debatable word, call Enchantment. Enchantment produces a Secondary World into which both designer and spectator can enter, to the satisfaction of their senses while they are inside; but in its purity it is artistic in desire and purpose. Magic produces, or pretends to produce, an alteration in the Primary World. It does not matter by whom it is said to be practised, fay or mortal, it remains distinct from the other two; it is not an art but a technique; its desire is power in this world, domination of things and wills.
It's not really clear from this description whether, for example, Galadriel's mirror falls under what Tolkien might call "Enchantment" or simply "Magic" proper. But neither, I think, describes anything that we see Legolas doing. He certainly has the ability to run quickly over snow without sinking or leaving much of a footprint; he is an extraordinarily good archer; he can walk a tightrope; he can see far distances with extraordinary clarity. These are all purely physical characteristics, though; there's no magic needed to produce those effects.
Other than that, we don't see him do much except follow and fight as he sees best.
Tauriel, of course, isn't present in any of the books, so there's no question of her doing anything; and Arwen has a much more passive role in the books (except, to an extent, in the "Tale of Aragorn and Arwen" in Appendix A)—I don't see any sense in which she uses magic either. We don't even know exactly the extent to which Elrond used magic in healing Frodo. We don't see exactly what goes on; though I suspect that the process was something like how Glorfindel began Frodo's healing:
He searched the wound on Frodo's shoulder with his fingers, and his face grew graver, as if what he learned disquieted him. But Frodo felt the chill lessen in his side and arm; a little warmth crept down from his shoulder to his hand, and the pain grew easier. The dusk of evening seemed to grow lighter about him, as if a cloud had been withdrawn. He saw his friends' faces more clearly again, and a measure of new hope and strength returned.
(Lord of the Rings, Book I, Chapter 12, "Flight to the Ford")