I think for me it's the first one. I saw it as more of a adventure film. I liked the introduction of the black riders (or is it dark ?) Seeing the sets such as the shire and the pub in that other town. The introduction of the main characters.
The FotR because they are out on a mission and meet everybody else along the way and RotK. If I have to choose I would go for FotR I guess. Because all is so pure and so new and suddenly their lifes change forever
Fellowship of the Ring is my favourite. Also, they took one of my favourite scenes from the Two Towers book and pasted it into the ending of that first movie, so generally I like it better for that reason too.
By far Return of the King. It is excellent, and, besides being a great fantasy film, it's one of the best films I've ever seen. Everything: acting, story, cinematography, special effects, is a work of art that really culminates into an awesome 3 hour experience. No wonder it won a ton of Academy Awards!
I can't say which LotR film is my favorite; I like all of them. Fellowship started it all and TT/RotK expanded upon and closed the story, respectively. They all had their spectacular scenes. However, I guess I'll give it to Fellowship of the Ring because of the distinct lack of Boromir in both Two Towers and Return of the King. I liked Boromir, a lot, even if he had a lust for the Ring.
I'd have to say Return of the King. I love all the battles and the ending makes me cry so much because it's happy and sad and just so much amazing rolled up into one movie. Plus that scene where Legolas takes down that Ophiliant thing is so awesome. The Rohirrim are really cool as well, pretty much everything in that movie is awesome. Can't even describe so much awesome in this post.
It's hard to choose one as a favorite, as I see them as one story that is told in 3 parts. I'd probably give the slight edge to FoTR tho. I'm probably not being terribly fair here however, as my nostalgia for the first film might be weighing my decision. Going to the theater and seeing that when it first came out is definitely one of my fonder movie going experiences.
Being more subjective I'd also say I enjoy the first part of an adventure more than the ending of one in most cases. Maybe it's because I know it's going to end when watching Return of the King and don't want it to so I go into it already a little dissapointed? That probably sounds rather silly, but it's how my brain works for whatever reason!
I do love all three movies (especially the expanded versions, because you get so much more character story and interaction blended in with the action - and I'm all about the characters). Personally I would have to name Two Towers as my favorite. Fellowship is by all means a good start, but it's a bit slow-moving compared to the other two and we are only just beginning to know the characters. Return of the King was a lot more action-packed movie in comparisson, so it didn't have the same character depth and focus as the previous two, not to mention the very dragged-out ending. Two Towers I felt was more a perfect blend between the two. We got a lot of moving and character-driven storytelling as the Fellowship entered Rohan (and as a Scandinavian, I can't help but love the clear Viking-vibe of the Rohan people), we also got a great battle at Helm's Deep and the battle itself was a lot more moving than the final battle due the way we got to actually know the people of Rohan prior to it (we didn't really get that with Gondor - there it was much more about the power-struggle and the tragedy of the steward and his family, the people itself got no real focus). So Two Towers is my clear favorite out of the three, but I love them all as they are all necessary to tell the full story