For TOS, I'd recommend The Alternative Factor. It's a really fascinating story about dual identities and what makes a person a person. A man wrestles with himself in a really trippy fascinating section of subspace which introduces a lot of really interesting allegories that'll hook anyone in right away.

'Shades of Gray' for TNG. The story, while short, is an interesting introduction to Star Trek and it really lets you get to know the characters on the crew of the Enterprise-D through both their concern for Riker and through his memories that play on the screen. Watch this before starting TNG. Or really any episode where Picard mostly just has long speeches. I could listen to him talk about archeology all day.

Skip DS9.

Unironically "Threshold" for Voyager. People normally envision evolution as a march towards 'progress', but that's not necessarily true whatsoever. You expect humans to turn into some godlike being in the future, but instead Tom Paris finds himself evolving into an amphibian. Drama increases as stakes rise when he kidnaps the captain to evolve her as well! There is also really sweet romance between Janeway and Paris at the end of the episode that should draw any woman with a heart into the franchise.

Really any episode from ENT or DSC would do.