Watch these 26 episodes and you can't go wrong They convey the essential TOS experience (and 26 is a lot less than 79).
Listed in broadcast order.
S1
The Naked Time
The Corbomite Maneuver
Balance of Terror
Arena
Tomorrow is Yesterday
The Return of the Archons
Space Seed
Devil in the Dark
Errand of Mercy
The City on the Edge of Forever
Operation Annihilate
S2
Amok Time
Mirror, Mirror
The Doomsday Machine
Journey to Babel
Obsession
Wolf in the Fold
The Trouble With Tribbles
The Immunity Syndrome
By Any Other Name
The Ultimate Computer
S3
The Enterprise Incident
Spectre of the Gun
Day of the Dove
The Tholian Web
Plato's Stepchildren
Send her to the snakes!

Most people do feel that the third season has a lot of subpar episodes, but that it does still have a few other good episodes although each person has their own idea of which those are.
Agree. In addition to The Enterprise Incident, two of my favorites from season 3 are The Tholian Web and Spectre of The Gun.
My top 3 episodes of the series (which have already been listed) are:
The Menagerie
City On The Edge Of Forever
Mirror Mirror
Al - Alicia
An - Andrew
Jo - Joseph
Be - Benjamin

sukhisoo has a pretty darn good list. I might just add a couple from the third season, as that season is the riskiest venture, and thus the most helpful with which to be guided:

• The
  Enterprise
  Incident (the other major encounter with Romulans after "Balance of Terror", and more or less a direct sequel as it centers on the same issue although nevertheless quite different in its story)
• Day of the Dove (the other major "personal" interaction with Klingons)
  Most people do feel that the third season has a lot of subpar episodes, but that it does still have a
  few
  other good episodes although each person has their own idea of which those are.

Sukhiso has a great list; I'm going to reinforce a few and add a few different ones:
The Corbomite Maneuver
Naked Time
Balance of Terror
Devil in the Dark
City on the Edge of Forever
Amok Time
The Trouble With Tribbles
The Doomsday Machine
Mirror, Mirror

The following episodes are usually considered the best and/or iconic:
Balance of Terror
Menagerie
Arena
Space Seed (which introduces Khan, an important villain in the films)
Errand of Mercy (Introduces the Klingons, who are present in most of the films)
The City on the Edge of Forever
Amok Time
The Doomsday Machine
The Trouble with Tribbles
Mirror, Mirror
Journey to Babel (which introduces Spock's parents who are featured in some of the films)
You may also want to watch Tomorrow is Yesterday, which describes a time travel technique used in one of the films.
Other lists may differ from mine.