Their are top 10 threads but I'm also interested in continuation stories followed up in the later series.
As far as episodes that are continued go...
"The Naked Time" has a sequel (more of a remake, really) on TNG.
"Errand of Mercy" has a prequel on ENT.
"Mirror Mirror" has several sequels on DS9.
"Journey to Babel" introduces Sarek, who reappears in several of the movies and TNG, as well as several aliens which are important to ENT.
"The Tholian Web" has a sequel on ENT (which is also a sequel to "Mirror, Mirror").
The Klingons from "Errand of Mercy," "The Trouble with Tribbles," and "Day of the Dove" reappear on DS9 (one is also on VOY).