Here's a good amount of stories to get a coherent idea about what Star Trek is about, in order watch:
Where no man has gone before" (the real first episode)
"The naked time" (it has a sequel in The Next Generation)
"The enemy within" (it's been parodied)
"Balance of terror" (introduces Romulans in a classic submarine style thriller)
"Day of the Dove" (good introduction to Klingons)
"Arena" (it's been referenced in Bill and Ted's Bogus Journey, among others)
"Space Seed" (Star Trek II is a sequel)
"City on the edge of forever" (award winning time travel drama)
"Amok Time" (explains Vulcan mating habits)
"Mirror mirror" (the alternate "evil" universe. You've seen it referenced)
"Trouble with tribbles" (funniest episode ever)
Star Trek II The Wrath of Khan (Star Trek the Motion Picture is unnecessary)
Star Trek III The Search for Spock (Direct sequel to II)
Star Trek IV (funny and accessible)
Star Trek the Next Generation: Encounter at Farpoint (the pilot)
"Q Who" (introduction of the Borg)
"Best of Both Worlds" (most consequential confrontation with the Borg)
It depends which characters from TOS and TNG you want to see more of after that.
I'd recommend the movies Generations and First Contact next.
Then, watch the pilots of Deep Space Nine ("Emissary"), Voyager ("Caretaker"), and Enterprise ("Broken Bow") to see if they interest you. Or watch the JJ Abrams movies if you've grown to love the Original Series crew but wouldn't mind new actors playing them.