I can't remember what scene stuck out the most the first time I read LoTR, because it was too long ago.
But always now, when rereading the story, the scene where Tom Bombadil puts the ring on and DOESN"T disappear, then actually makes the RING disappear always freaks me out.
After the first reading, and knowing what a "big deal" the ring is, Tom's nonchalance gives me goosebumps. The "rules" are being broken!