[Kransha]

Well, Elrond and Arwen are Half-Elves, not full Elves, so slightly different. His lineage is rather complex, as is Arwen's.

Elrond's dad, Eärendil, was Half-Elven, the son of a Noldorin Elf (like Galadriel) and a regular mortal man. Eärendil passed this on to his two sons, Elrond and Elros (Elros chose to be mortal, and died, but not before starting the Line of the Númenorean Kings, of which Aragorn Elessar is a part). Here's a slightly more accurate portrayal of Elrond's lineage, ala family tree.
-He was part Sindar, because his grandmother, Nimloth, and great-grandfather, Thingol, were both Sindarin Elves. This makes him about 5/16ths Sindar, all in all, from Nimloth and Thingol Greycloak.
-He was part Maia (like Gandalf) through his great-great-grandma, Melian, Thingol's wife. This is quite an honor, but Elrond is only 1/16th Maiar, with barely any of the inherited blood from Melian still in him.
-He was part Noldor, for the reasons mentioned above. He is about 3/32nds Noldor, with the addition of his great-grandfather Turgon, who wasn't wholly Noldor anyway.
-He was part Vanyar, through his great-grandmother Elenwë AND his triple-great-grandmother, Indis, the second spouse of Finwë, the Noldor Lord. He's all of 5/32nds Noldor, in blood and mind.
-And, even though he's a Half-Elf, he's really only 3/8ths Mortal (or Edain), through Tuor, his grandfather (House of Hador), and Beren, his great grandfather (House of Bëor).

Gets a bit confusing, don't it?

Legolas would speak Sindarin, but with a dialectual twist that would make it "Silvan," which is similar. In LotR, the Mirkwood Elves only speak in Sindarin, with a Silvan accent, because basic Silvan went out of style. Tolkien tells us:
"By the end of the Third Age the Silvan tongues had probably ceased to be spoken in the two regions that had importance at the time of the War of the Ring: Lórien and the realm of Thranduil in northern Mirkwood. All that survived of them in the records was a few words and several names of persons or places."
-Appendix A, The History of Galadriel and Celeborn, UT

...That really solves the problem pretty well.

P.S. You're right, Sindarin technically is a race, but not in comparison to Quenya. Quenya and Sindarin are languages, and Sindarin is not exclusively the language of the Sindar Elves. The Sindar are Telerin, I believe, and dwelled in Beleriand during the First Age of the Sun.