I think the first one was Stand by Me. I watched it as a little kid and It was all so dangerous and yet they mostly didn't seem any braver than I was. They each acted so fully and nuanced. Richard Dreyfuss' ending narration sealed the tears.
but the first movie to completely overwhelm me with tears was Dead Poets Society. My face was drenched and I felt like Ethan Hawkes character completely bereft and angry and devastated. To this day, when I hear the score, I start to tear up, if not fully cry.