Although Tolkien's writings on Quenya and Sindarin are extensive, there simply isn't enough to carry on realistic conversations; unless you're willing to go into some of the fan attempts to flesh the language out, you and your Elvish-speaking partner are basically limited to quoting passages of the Legendarium at each other.
Tolkien himself actually acknowledged this in Letter 297 (emphasis his):
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.
Also of interest to some, and agreeable to me, would be an historical grammar of Quenya and Sindarin and a fairly extensive etymological vocabulary of these languages of course far from 'complete', but not limited to words found in the tales. But I do not intend to engage in these projects, until my mythology and legends are completed.
The Letters of J.R.R. Tolkien 297: To 'Mr. Rang' (draft). August 1967
Klingon, on the other hand, is quite a bit more extensively-developed, to the point where you actually can read (and perform) Shakespeare "in the original Klingon."
John McWhorter, professor of linguistics (among other things) at Columbia University, discusses fictional constructed languages ("conlangs") in this Ted-Ed video, and comes to the same conclusion:
Relevant transcript (emphasis mine):
McWhorter: The truth is, though, that Elvish is a sketch for a real language than a whole one. For Tolkien, Elvish was a hobby rather than an attempt to create something people could actually speak. Much of the Elvish that characters in the Lord of the Rings movies speak has been made up since Tolkien by dedicated fans of Elvish, based on guesses as to what Tolkien would have constructed. That's the best we can do for Elvish, because there are no actual Elves around to speak it for us. But the modern conlangs go further: Dothraki, Na'vi, and Klingon are developed enough that you can actually speak them.