zach8270
Mugetsu Ookamiza
I typically suggest that people start with anime from genres that reflect the kind of stuff they'd look for in other forms of media. it's worked to help me get people I know into anime (either to keep them from talking about me "wasting time and money" on it or because they were curious about it). so I typically like to know what kinds of TV series and movies people watch and what kinds of books they prefer before giving recommendations.
Therein lies some of the problem. I don't really know where to begin looking for that. My tastes are all over the place. Mythology, zombies, horror, sci-fi, giant robots, explosions... and sometimes the just plain odd and goofy. That's why I felt myself intrigued but confused by FLCL.
that can make it kind of hard to start out, but I think I can give at least some suggestions
Evangelion (sci-fi, giant robots, and all that and references to Judeochristian mythology and mysticism)
Coyote Ragtime Show (sci-fi, explosions)
Full Metal Panic (sci-fi, giant robots, and lots of fun, especially if you watch Fumoffu, which was released between season one and season two proper)
Boogiepop Phantom (horror, mystery)
Master of Mosquiton (horror/comedy, an older series but it's an intentionally corny humorous romp with horror aspects, fun to watch with others or while drinking if you're old enough to)
Shinesman (an amusingly odd parody of anime and sentai series)
Scrapped Princess (sci-fi masquerading as fantasy, highly underrated but most of the people I know who watched it loved it)
and if you have any interest in fantasy, then Moribito, Twelve Kingdoms, and Kemono no Souja Erin are great watches (don't let the last one fool you if you decide to check it out, it seems really childish at first, but about 7 eps in you get a wham ep that shows you there's more to it than what there first seems to be).
I could probably come up with a few more ideas, but too many may give you too many to chose from.