My Neighbor Totoro (1986)
Showcasing the films of Miyazaki to my roommate, this is a film that works strangely enough because it is a kids movie. I like the mostly realistic storytelling. It's pretty much two kids moving in to a new place, with only a couple magical things happening on occasion. A nice, warm G-rated film for the whole family. Not often that I say that as a compliment, but it works here.
Spirited Away (2002)
I'm not sure I can call this my favorite Studio Ghibli film like most people seem to. It's too episodic for my tastes. It's still a good film, and I recommend it for the animation and technical stuff alone. If that isn't convincing enough, it is a very imaginative film, and it wouldn't surprise me if the appeal was in the creation of all these no-face's, good and evil twins, and giant babies.
Castle in the Sky (1986)
I guess if I have to choose favorites, this is mine. This one's a lot of fun, and I love how gritty this feels compared to even some of Miyazaki's PG-13 rated films. This is a film where kids almost die. A lot. I'd call it Indiana Jones for the anime crowd, but I don't think that does it justice. The film has some surprisingly deep themes about nature and power, and it features a robot that can easily put the terminator to shame.
Note: For the curious, I watched all three of these in their original Japanese with English subtitles. I'm not a fan of Disney's attempts at English translation, and hiring two grown adults to voice children in Castle in the Sky is an insult to those who want plausibility.