I like a TV show and a film that's interesting to watch, reasonably well paced (I have a lowish boredom factor when it comes to passive entertainment), and is sufficiently deep that there's something for my brain to dig into.
I'd tend to suggest Paranoia Agent.
Try to watch a fansub of it. It's a 13-episode show that could be a good entry point into anime. It's a thriller about a roller-skating boy with a cap who assaults random people with a baseball bat, and people are trying to figure out who he is. It's both weird and deep. Especially from episode 5 on. Has some surrealistic stuff, opportunities to ponder over what's being shown versus what you think you've figured out, etc... I watched the entire show twice and even the second time, I didn't figure it all out. Well, I suppose it's mostly targeted to those who like The Prisoner, things like that ;)
From the same director (who sadly passed away last year from cancer) -- Millennium Actress is an excellent 'romance' movie about an old actress who's reliving her memories under the prism of her major movie roles -- they're intertwined so much that you're not sure whether you're watching reality, or an idealized version of it. And Tokyo Godfathers is a Christmas story about three bums who discover an abandoned baby and decide to take care of her. It's a character-driven comedy/drama. Mostly comedy.
Of course there's always the Ghibli movies to start with... If you want a sense of adventure, there's Laputa (Castle in the sky), or a sense of wonder, go for Kiki's Delivery Service or Howl's Moving Castle. If you want drama, there's Nausicaä or Princess Mononoke. Ghibli's most popular movie is Spirited Away but I'm not a big fan myself. If you like tragedies, best of all is Grave of the Fireflies.
Also in the 'weird surrealistic' area, I could think of Utena, but the main drawback here is that out of the 39 episodes, most are of the 'formulaic' kind, i.e. it's always the same structure, character has a problem, it's analyzed in the first half, then they get to fight, and their problem is solved (or not). Plus it has a soundtrack that could be seen as either genius or pure crap. (I personally love it but I can understand it's not something everyone will like.) Hmm... Probably not a good entry point.
One of my favorite movies is Princess Arete, if Liz ever wants to watch a slightly feminist movie, and you both don't mind with the very quiet pace, it's really great.
Actually, Milady hates Japanese animation in general (with a passion!), but over the years she liked all of the anime I mentioned above (except for Utena which I wouldn't dare to even introduce her to.)
Oh... If you like over-the-top animation/story, you may enjoy Giant Robo: The Animation (1993 or so). Yeah there are giant robots in it, but it's a cult show that pays homage to the old Japanese 60's shows. A labour of love, really. I particularly recommend episodes 4, 5 and 7 (mostly because the first episodes are mostly exposition.)
And Evangelion (1996)... Because it doesn't deliberately choose a retro style, it looks kind of dated by now. They recently made new movies out of it which are high budget and everything, but you may want to wait for a couple more years until they're done with the remakes. Otherwise, well... It's another giant robot show with kid pilots, but this time it's on a Freudian level... Guy has a complex around his father, father is obsessed about reviving his dead wife or something like that, mix it up with tons of esoteric references, trauma, and a share of typically Japanese gags. It's kind of... special. If you manage to get through the first half of the show, which is quite commercial and a bit boring, the second half rewards anyone with a brain. It's basically the first prime-time anime series to show an orgasm off-screen (in episode 20), or have an episode entirely built on intertitles and still images (the very controversial episode 25).
Oh, and Cowboy Bebop... A pretty good jazz/funk soundtrack, stand-alone episodes that allow for casual viewing, mostly a comedy show with action scenes. I'm not a big fan (except of the very Batman TAS-inspired episode 20) but it's one of the best loved shows of its kind.
Music... really, the music in TV and film doesn't tend to grab me unless it's *seriously memorable* or *seriously epic*. I don't own many film soundtracks, for example. To put in perspective, the film soundtracks I own are: Indiana Jones & The Last Crusade, Flash Gordon, the three Lord of the Rings soundtrack albums, Repo! The Genetic Opera, Labyrinth, and Clerks II.
I have almost more computer game soundtracks than I do film soundtracks.
And still looking... Anything that sounds like the 16-bit era is what I'm seeking after. Not the 8-bit era though... Except if it sounds like a Commodore 64, in which case it's okay :P
That said, I do have the soundtracks from the first 5 seasons of Doctor Who, because somehow that's actually more memorable to me.
It's mostly because they don't do a lot for me, as it were. If the dialogue's good, that's going to keep my attention more than the music is.
Although I watched 'The Sky Crawlers' this morning (from Mamoru Oshii), and I found it to be really well written. Only problem is, the story wasn't so great... (Well, I'm not a big fan of depressing movies with no encouraging ending :P) One of Kenji Kawai's best soundtracks though (along with Patlabor 2), if you like minimalism. I have a feeling you don't, though!
Musically I tend to favour rock, not-so-heavy metal and stuff like that. My playlist is primarily Queen, Meat Loaf, Iron Maiden, Nightwish and Murray Gold(!) with interludes from Bon Jovi, ABBA(!!), David Bowie, Aerosmith and stuff like that, plus whatever I happen to find of interest on YouTube.
ABBA is great, just like A-HA. Gotta love old Scandinavian pop. Anyone who says otherwise is lying! Ahah.
