Galadriel was perhaps the only known High Elf in Middle-earth during that time who (having being born in Valinor) had learned Quenya as a cradle-tongue. The poem Namárië is the longest piece of Quenya found in The Lord of the Rings. It is also known as Galadriel's Lament.
("A star shines on the hour of our meeting.") Galadriel is perhaps the only major Elf character in The Lord of the Rings that had learned Quenya as a cradle-tongue: she was born in Valinor in the days of the Two Trees.
Of these two groups of Elves, most of the Noldor returned to Middle-earth where they met the Sindarin-speaking Grey-elves. The Noldor eventually adopted Sindarin and used Quenya primarily as a ritual or poetic language, whereas the Vanyar who stayed behind in Eldamar retained the use of Quenya.
Her name "Ar-wen" means 'noble maiden' in Sindarin. She was given the name "Evenstar" as the most beautiful of the last generation of High Elves in Middle-earth. Arwen said: "Dark is the Shadow, and yet my heart rejoices; for you, Estel, shall be among the great whose valour will destroy it."
As an elf of Mirkwood, Legolas speaks Silvan, or least knows some words and phrases. Finally, he definitely speaks Sindarin, which is the most common elvish language at the time of the quest to destroy the One Ring.
However, as Elrond (and, of course, Elros) were raised by Maglor, but were born in Middle-Earth, would they speak Sindarin or Quenya? They might have spoken both. Having been born at the Havens of Sirion, they most likely spoke Sindarin as their birth tongue.
As others have noted, 'Galadriel' is a Sindarin name, not a Quenya name. It means 'radiance-crown-girl', or 'maiden crowned with a garland of bright radiance' if you want to be more poetic.
Legolas is a Sindarin Elf from the Woodland Realm of Northern Mirkwood. His father, Thranduil, is the King of the Silvan Elves living in that realm, making Legolas the Prince of Mirkwood.
Most of it is Sindarin, because it's the elvish lingua franca in western Middle-earth (and also spoken in parts of Gondor); you'll notice the rare Quenya (like in Galadriel's Namarie song) because of how different it looks and sounds.
She was a royal Elf of both the Noldor and the Teleri, being a grandchild of both King Finwë and King Olwë. She was also close kin of King Ingwë of the Vanyar through her grandmother Indis. Galadriel was a leader during the rebellion of the Noldor, and present in their flight from Valinor during the First Age.
Sindarin would be the language to use when naming any Elf of Middle-earth in the Second Age and beyond. The TL;DR version of all that waffle? Quenya is limited in its usefulness to a few specific situations. Looking through all of Tolkien's writing, the vast majority of names that appear are Sindarin.
Which one should learn first? This question can't be answered directly: it largely depends on what languages you're already familiar with. I believe that Quenya is appropriate for the starter because it gives better understanding of the Primitive language, and then he can find Sindarin easier.
This was later called Quenya (High-elven) and is one of the two most complete of Tolkien's languages (the other being Sindarin, or Grey-elven). The phonology and grammar of Quenya are influenced by Finnish, while Sindarin is influenced by Welsh.
In Gondor at the end of the Third Age, Sindarin was still spoken daily by a few noble Men in the city Minas Tirith. Aragorn, raised in the safety of the Elvish stronghold of Rivendell, spoke it fluently.
Over the years, Gandalf became his most common name, given to him by the humans of Arnor. However, Mithrandir was another popular name for Gandalf in The Lord of the Rings. This was Gandalf's Sindarin name, used by the Elves and later the citizens of Gondor.
Thingol learns of the Kinslaying at Alqualondë. He bans the speaking of Quenya of Beleriand; the Noldor adopt Sindarin and only speak Quenya in private and in Gondolin.
Galadriel (then called Artanis and Nerwen) was born in Valinor during the Years of the Trees. Therefore, her first language was Quenya—the language of Valinor, that even the Maiar and Valar used (though among themselves, they preferred Valarin).
Arwen means "Noble Maiden" in Sindarin, from ara- ("noble") and gwenn ("maiden"). Tolkien remarked that it also means "greatly blessed" in Welsh. The Quenya form of her name is not entirely certain, but in his Quenya greeting, Aragorn refers to her again as "Arwen" (Arwen vanimelda, namárië!).
Tolkien intended the name Gondor to be Sindarin for "Stone-land". This is echoed in the text of The Lord of the Rings by the name for Gondor among the Rohirrim, Stoningland. Tolkien's early writings suggest that this was a reference to the highly developed masonry of Gondorians in contrast to their rustic neighbours.
The name Morgoth is Sindarin (one of Tolkien's invented languages) and means "Dark Enemy" or "Black Foe". Bauglir is also Sindarin, meaning "Tyrant" or "Oppressor". "Morgoth Bauglir" is thus an epithet.
Aragorn was called Estel (Sindarin for "Hope") instead. During his life in Rivendell he accompanied the twin sons of Elrond, Elrohir and Elladan, on their journeys. "Estel" grew to be fair and noble, and was seemingly more mature than his age.
