As a TOS fan here are the ones that I'd absolutely watch, in this order:
Season One
"The Cage" un-aired pilot, since it sets up SNW.
"Where No Man Has Gone Before", the actual TOS pilot.
"The Corbomite Maneuver", since it was produced next and introduces the rest of the main characters except Chekov
"The Man Trap"
"Balance of Terror"
"The Conscience of the King"
"The Galileo Seven"
"Court Martial"
"The Menagerie" two-parter, uses "The Cage" footage but you need the other episodes first
"Shore Leave"
"The Squire of Gothos"
"Arena"
"A Taste of Armageddon"
"Space Seed", sets up the second film and every plot point to do with Augments in all of Trek
"The Devil in the Dark"
"Errand of Mercy"
"The City on the Edge of Forever"
Season Two
"Metamorphosis", introduces Zefram Cochrane the inventor of warp drive
"Friday's Child"
"Amok Time" for a spot of Vulcan world-building and the best/cheesiest/best and cheesiest fight scene in all of Trek
"The Doomsday Machine"
"Mirror Mirror"
"The Trouble with Tribbles"
"Journey to Babel"
"Obsession"
"The Immunity Syndrome"
"A Piece of the Action"
"The Ultimate Computer"
"Assignment: Earth"
Season Three
"Spectre of the Gun"
"The Enterprise Incident"
[You could watch "Spock's Brain" here simply for the camp value
]
"The Tholian Web"
"For the World is Hollow and I Have Touched the Sky"
"Let That Be Your Last Battlefield"
"Whom Gods Destroy"
"The Lights of Zetar"
"All Our Yesterdays"