Ghibli for sure. I got my mother to watch a few as I knew she'ld like them [She eventually went out of her way to buy her own "Cat Returns" haha] and she's hooked on them, always excited when I tell her about new ones, haha.
In turn, she asked to watch some series, so i showed her Bleach, Soul Eater & D-Gray man, as its not hard for someone who's not used to Anime to enjoy them ..and yeah, she liked them
My brother really enjoys Eureka 7, even though he's not really into Anime so I think Mecha/Action is a good thing for the majority of guys to start on.
My personal gateway was [excluding what I saw as a kid as my interest didn't flower till till I was 18-19] Ghibli, saw Totoro in the shops, thought it looked cute, so I bought it and BOOM, hooked.
EDIT: Can I just say my mother is awesome? Open-minded about all my interests, <3 her. She even giggled at P&S when she glanced over my shoulder & Thought the Elfen Lied opening was "beautiful" :O Rather than "OMG, ANIME IS ALLLZZZ PORRNNZZ"
Sorry for the off topic-ness haha.