Being a huge star trek fan, ive seen pretty much every series mutliple times (NuTrek excluded). Ive never watched all of TOS. I dont know why, but the 60s vibe puts me off everytime i start it.
If you like, you can start off with the serious eps, which have less 60s nonsense: Balance of terror, City on the edge of forever, The enterprise incident, The doomsday machine, and my favourite, an episode that could be used to define Star Trek: Devil in the dark.
After that there's a bunch of goofy stuff: going to the Nazi planet or the mob planet or the Roman planet, tribbles, Shore Leave, Mirror, Mirror, and some that is a mix of goofy and serious: Corbomite Maneuver, Arena (the Gorn episode), Galileo seven, and a bunch of others I've forgotten.
The one part where TOS is comfortably clear of TNG is the movies: I love 1 though it divides opinion, 2 is a classic, 3 is good, 4 is the sweetest sci-fi movie ever made, 5 sucks, and 6 is good again. Would need to watch a little bit of the TV show to get the character dynamics before starting the movies.