Depends what you are looking for as anime is vast.
It also depends how much does reading subtitles ruin the experience for you. If cutesy is harmful or not. Do you want action? Suspense? Gore?
Full Moon Wo Sagashite for example while is my favorite anime isn't one I'd start someone with. Given its cutesyness, its 64 episode run, and that you have to watch the fansubs because the official dubs and subs are incompetent.
Perhaps... Prince of Tennis may be better for you.
By Prince of Tennis, he means the Live Action musical. That is DE-licious.
But yeah, it really depends on what you're looking for in a show, as anime spans the entire gamut of tastes and genres. Give us just a tiny bone, just a nibble, and we'll pull something amazing out of our big bag just for you.
Yes, I should have realized the breadth of possible answers to my question was too vast. I think perhaps I can only offer vagaries, but I shall do what I can.
1. I really do not have too much stomach for “cutesy”, I must confess.
2. Reading subtitles would degrade the experience too far.
3. I would prefer something suspenseful or mysterious, but note that I have no taste for horror—I spend too much time alone with my imagination for that!
4. Well-dressed people are not a requirement but a boon. That would be “well-dressed” by my eccentric standards—good hats, frock coats, spectacles, boots, and the like. I suppose it is a rather strange thing to add, but I have not too much else.
5. Anything cerebral would please me—something that might give my mind something to chew.
This probably is a naïve, vacuous list, but I know nothing of anime, and I rarely watch any kind of entertainment at all, so it is hard to speak of my tastes. I chose this anime topic because I think I would like to watch
something in addition to the occasional
The Twilight Zone episode, and this seemed as good a place to start as any.