It is not possible to speak Tolkien's Elvish Languages.
This may confuse some people, considering how much nonsense there is online for "how to speak Tolkien's Elvish" (there's even a wikihow article about it). But according to the Elvish Linguistic Fellowship, an organization dedicated to the study of Tolkien's invented languages, even Quenya and Sindarin are simply not learnable:
The vocabulary, grammar, and syntax of Tolkien's invented languages, even of Quenya and Sindarin, are far too incomplete to allow its casual, conversational, or quotidian use. As Tolkien himself stated, "It should be obvious that if it is possible to compose fragments of verse in Quenya and Sindarin, those languages (and their relations one to another) must have reached a fairly high degree of organization — though of course, far from completeness, either in vocabulary, or in idiom" (Letters p. 380). Indeed, it was never Tolkien's intent to make Quenya, Sindarin, or any of his languages into spoken, written, auxiliary, or otherwise "useful" forms; rather, they were done for purely personal enjoyment. As Tolkien wrote, "It must be emphasized that this process of invention was/is a private enterprise undertaken to give pleasure to myself by giving expression to my personal linguistic 'aesthetic' or taste and its fluctuations" (ibid.)
-- the aforementioned E.L.F.'s FAQ, emphasis mine
An article entitled "Elvish As She Is Spoke" goes into more depth about the history of Tolkien's Elvish and the problems with trying to learn to speak it.
"But don't they speak Elvish in the films?"
The lines and lyrics from Tolkien's languages in the Peter Jackson films were written by David Salo, a linguist who wrote a grammar of Sindarin. He built the dialogue he needed on the limited available attested vocabulary, coining some new words himself but with the goal of making it intelligible to any viewers familiar with Tolkien's languages. To quote him:
Why is there Elvish in the movie? Why did Peter Jackson care enough to strive for some accuracy in the way language is presented? (...) The Elvish in the movie is addressed to the minority of viewers who know something about the languages. And what are they going to want to do when they hear the Elvish sentences? They're going to want to figure out what they mean, and why they mean what they mean. Part of my intention, my particular vision and contribution to this movie, was to create sentences which would be intelligible to the people who study the languages (...) I'm enormously happy to see some people saying based on their knowledge of Elvish, great or small, that they recognized and understood some of what they heard on the screen. That's great - that's exactly the kind of effect that I was looking for.
Here is a collection of the translated dialogue from the LoTR films, with the attested and unattested vocabulary in each line listed. As you can see, a lot of it is attested in Tolkien's works, but there is quite a bit that needed to be invented for the film by Salo. There have been other projects based on Tolkien's Elvish languages in the past as well, but the real fanatics tend to view these as "pseudo-Elvish", since anything not invented by Tolkien can't really be said to be part of Tolkien's Elvish. That said, there's nothing wrong with seeking out resources for and learning these expansions on Tolkien's conlangs -- just know that calling any one an authentic recreation of Tolkien's Elvish is pretty flawed.
Since Tolkien never fixed his languages firmly or described them completely enough to provide any such comprehensive and corrective model (that never being his goal), ... it is consequently a further inescapable fact that no one has or ever will be able to speak Quenya and Sindarin, any more than anyone will ever (again) be able to speak, say, Etruscan or any other fragmentarily-attested non-living language.
Look into the E.L.F.'s resources (though they do seem a tad outdated) if you're interested in further exploring what information we have about Tolkien's Elvish.