Howdy and welcome, Nólatári (I can just about figure out what that means; yea!) Looks like perhaps Celebrian has found her fellow sixteen year old, and didn't take long at all. The movies, don't get me started on the movies. Blonde Quendi everywhere, fie! Everything else I might could forgive, but Frodo sending Sam home from Mordor was the anvil that broke the camels back. You don't just turn around and go home from Cirith Ungol, and besides, Frodo would never have done that. The unbreakable friendship of Sam and Frodo is one of the great themes and great beauties of the Trilogy; even the Ring could only slightly damage, but never destroy it.
And yes, the Trilogy is just the tip of the iceberg. I really need to read it again since it's been so long, but my heart will always belong to the Silmarillion. You can enjoy the Trilogy without it, no question, but it improves even more on so many levels in the context of the Silmarillion (in which the Trilogy gets about thirty pages.) The Trilogy heroes are unquestionably great, but their stature simply can't compare to those of the Elder Days Yet again I say Feanor would've made short work of Sauron (heck, Finrod Felagund almost took him. I personally think Aragorn is the equal of Beren or the other Men of the First Age, but don't ask me to prove it.
The bittersweet sorrow only alluded to in the Trilogy of the Eldar, much "diminished," to use Galadriels word, from their grandeur in the First Age, of a world that was wholly theirs now witnessing the last of their greatness before those last Eldar depart forever is far more keenly felt and shared by the reader with a familiarity with the Silmarillion. Some may (and do) debate the merits of History of Middle-Earth, but the Silmarillion is a must. Otherwise it's like learning the American Revolution by studying Yorktown.
Of course, you may be including that when you call the Trilogy the tip of the iceberg (I've found the level of dedication/fanaticism at this site much higher than I've encountered anywhere else; I fit in, which is disturbing) in which case I'll be very red in the face. However, if not, and you've wondered about Gil-Galad, Earendil, and Varda (to name but a few; I'm not gonna cover the whole Quenta Silmarillionin a post) there's your source. I lament the lack of Trilogy level detail, but Tolkien never finished it, despite working on if for almost sixty years. It's comprehensive, but leaves me wanting more (perhaps inevitably.) For me, that's where History of Middle-Earth comes in, but that's confusing enough if you HAVE read the Silmarillion; it would be impossible without it.
Again, welcome and enjoy your stay.