Well... I watched the FOTR before I read the book. I had no idea what it was all about. You can say I saw it by accident - my friend's brother had one spare ticket and I got invited.
At first I was amazed by the creativity of it. After a couple of minutes I knew I have to read the book! I didn't know anything about the story, so I was trying to learn as fast as I can, but of course - not to long after the opening scene I was pretty much lost Gandalf was the first character, who caught my attention - I was scared when he was prisoned in Isengard, and I was very heart broken when he fell in Moria. I was so confused and troubled by it, that everything after that scene was quite blurry for me - after the movie I didn't even remember what happened after Moria sequence. And of course - I didn't understand the ending - I thought "Well, that's a strange ending!" and said that to my friend and her brother (both of them read the books long before I did) and they said tht it wasn't the end at all. I remember I felt quite stupid then. They also told me that Gandalf didn't die. I had some questions of course, so they explained the plot a bit for me and then I read the books - I actually devoured them, long before The Two Towers were in the cinema.
Of course after the movie I was overwelmed by it's beauty, it's epicness, scale and depth of the whole story (even if I didn't know anything about the Middle Earth at that time). I loved Ian McKellen's character, I still do - it's probably my favourite character in Tolkien's books till this day. I wasn't sure about Lady Galadriel being the most beautiful, the fairest creature in the movie (wasn't she supposed to be this way?). I loved the music, the sudden silence when Gandalf falls still makes the same impression on me. I loved the sceneries - I wasn't paying attention if they were real or CGI generated - they looked great. I wasn't a fan of Legolas character (due to the actor I think), and CGI effects in action scenes (Legolas and cave troll scene in Moria for example).