If you’re looking for an entry-level modern Dr. Who episode, it’s hard to do better than Blink (aka the first weeping angel one). Pretty good story, excellent villains, and you’ll never casually walk by a cemetery again.
Although one of my children may or may not be named after a character from DS9, I never really liked the series as much as the other ST series. Mostly, it’s because of Sisko, who I just never liked. OK, Enterprise was worse, but mostly because it went all-in on the post-9/11 plot line. Voyager did suffer from too many new particle shows (the ship has been saved by our new theta radiation generators that we’ll never see again!), but I liked most of the characters enough to overlook the flaws.