My Story is very similar to Broomie's brothers and all. The first game I remember playing was King's Quest V, I'm sure there were some before that but I very vividly remember just wandering about in the woods and then the witch would come and turn graham into a frog. It terrified me so badly, I would hide under the desk trying to click restore blindly because the image of frog graham staring down the camera was too scary. it was around 15 years later that I beat that game.
The first adventure game I finished was either Full Throttle or Curse of Monkey Island. They were both so easy compared to Sierra games that they made for good starters. When people ask about starting adventure games those two or The Dig are always my recommendation.
What makes them so good is you can play them like interactive stories. I've played Full Throttle start to finish (having a save to skip that weird combat bit) in front of two friends separately just to have them watch it. I played Curse of Monkey Island straight through in two sittings with my niece (11) and she said it was her "Favourite cartoon ever" and she really like how Guybrush was relatively unfunny in a world of crazy characters.
I played The Dig through with my fiance and her thoughts after praising it for hours were "Why isn't this in theatres right now?" now I've got all of those people hooked into my telltale account, because they're also easier games with focus on story and jokes, and I just light up when I get the occasional email about "Oh man, when Homestar races the hurdle the turtle and curdle....d milk it's so good" or, "This New Monkey Island is as funny as the ones you showed me."