Inuyasha is one of the essential anime series. It certainly is the one that got me into the genre. Inuyasha is about a teenage girl, Kagome, who discovers she’s the reincarnation of a powerful priestess when a demon attacks her ad she falls through an old well at a shrine and ends up in the past. Instinct leads her to find the titular character, a half dog demon-half human, who loved Kagome’s past self (with tragic consequences) and is destined to protect her. When Kagome accidentally shatters a powerful magical gem, she and Inuyasha must hunt the pieces down, collecting an engaging, motley crew of cursed, charming, and tragedy-afflicted sidekicks. Except, far from secondary characters, Shippo, Miroku, and Sango’s stories are tightly entwined with Inuyasha and Kagome’s own.
While Inuyasha does face some tragedy (demonic control, murder, monsters in human form, etc) and romance, the graphicness remains mild, and the romance is ultimately on the sweet, crushy type. Inuyasha himself is quite bratty (and Miroku is a lech, but it is mostly implied, and the other characters call him on his behavior all the time), all the characters are undoubtedly heroic, good, supportive people. There’s even a range of demons, from relatively good demons, to beneficial nature spirits to monsters and plotting evil overlords.
I highly recommend this one for most audiences.
Note: There are bare, nippless breasts on a demon in the first few episodes, but there is no other noticeable fan service.