I'd say, note what your family and friends like to watch, then pick out a few interesting titles that aligns with their tastes and have them watch an episode or two with you.
@mnmike pretty much has similar ideas that I would suggest.
I think it depends on the person. For someone younger, especially if they are male, something like Naruto or Hunter x Hunter is a good choice.
For young-to-middle age adults, I think starting with a Miyazaki film (I'd recommend Spirited Away) and then branching out into something a bit more serious (Full Metal Alchemist: Brotherhood, Toradora, Clannad, Trigun, Serial Experiments Lain, etc.) that fits whatever genres they like the best. I'd stay away from fan-service heavy shows at first--unless you know for a fact that they really like that sort of thing.
For older people, especially for those who are a bit more conservative, you can't go wrong with the film Tokyo Godfathers or the series Bunny Drop. I've shown them to both my mother and my in-laws, with very good results.