Here's the hard part: it's mostly cartoon for teenagers and some shows are for young adults and a handful have a broad appeal. If you took a random anime off the air, chances are it will be a lot more mature than stuff from the animation ghetto as it will have sexual motifs and wacky stuff that teenagers (who normally are long ago bored of simple cartoons except for Gendy Tartakovsky's self-satirizing works) are still quite interested in and can relate to.
So most anime would be more interesting to grownups, but the fact that it's still for a different demographic and from a different culture is a great barrier.
...however if you want to "convince" your peers or elders that your hobby is worthwhile than hunt for some titles that they could take a look at. Look for things with broad appeal and solid themes like Cowboy Bebop if they like movies and "get" pop-culture or Michiko to Hatchin if they like a solid road movie or Area 88 if you're more into old-school stuff and classical "manly drama".
(Chances are showing some deep, philosophical or wacky sci-fi stuff to your mom and pops will only alienate them as they can't relate to that exotic stuff... but I digress. If your da is an old-school sci-fi nut pop in Ghost in The Shell or Akira and you're home free... or some hard-core '80 sci-fi if he was more into Heinlein than Asimov).
I'd leave slice of life or romantic comedies last, as they focus on a slow subject (often put to macabre by soap operas) and the peculiarities of the culture they're set in really come through so it definitely wouldn't work as a gateway series for people already struggling with several barriers. (Prime barrier that cartoons are for kids).