There were three really instrumental movies for me, and I saw them all around the ages of 13-14:
Taxi Driver was the movie that got me into movies and indirectly made me aware of "film history." I heard some people talking about it on like, a talk show or something, and I noticed my dad owned it on DVD, so I just popped it in one day. I had never really gone out of my way to watch an older movie and I loved it. Also on the back of the DVD case it mentioned the AFI Top 100, so I looked that up and started watching those. So Taxi Driver was the movie that made me aware of the fact that there were cool older movies I could be seeking out, and not just going to see whatever was in theaters.
Once Upon A Time In The West was the movie that made me want to make movies. I caught it randomly on TV one day and it was the first time I found myself consciously paying attention to matters of technique and style... i.e. not just paying attention to what's happening in the story but considering "why did the director film it this way?" I felt after that movie that I wanted to be Sergio Leone.
Blue Velvet was the movie that opened my eyes to the emotional potential of the medium the most. Again I was like 14 at this point, and I had just never seen anything so raw and honest as that movie. I didn't know you could tell a story so uncomfortable and personal. I just didn't know movies like that even existed. Blue Velvet felt to me the same way that paintings and poetry did, like a pure unfiltered expression of one person's soul. It is still to this day the movie that inspires me the most. Whenever the film industry gets me down and I start to feel fatigued and discouraged about the idea of ever making a movie, I remember Blue Velvet exists and that it is possible to make something true and good in this medium.