One Piece (1999)
Okay, I know this is a show about pirates traveling on sailing ships, but hear me out: Even if we ignore the fact that Star Trek often draws parallels between space travel and sailing on the open seas, One Piece, by Eiichiro Oda, has both structural and thematic similarities to Star Trek that cannot be ignored.
Structurally, One Piece follows Monkey D. Luffy and his crew of Straw Hat Pirates, as they voyage to various islands. On each island, the crew is faced with a conundrum. Oftentimes, these conundrums can have serious ethical implications. Even if part of a story arc is spent with action and fighting, the crew still has to find some kind of solution to an island's problems, even if it's an imperfect one.
This, along with the sense of freedom that Luffy and his crew experience sailing the seas, gives One Piece a lot in common with Star Trek, most notably The Original Series. Plus, with over 1,100 episodes, One Piece has a kinship to Star Trek in terms of the sheer amount of material that's out there to watch.
(If the length is intimidating, however, there is a new adaptation of the still-ongoing manga in the works, which may be more manageable. There is also a faithful live-action adaptation of the first saga currently on Netflix.)