Season 1:

The Corbomite Maneuver Balance of Terror What are Little Girls Made Of? Court Martial The Menagerie Shore Leave The Squire of Gothos Arena Tomorrow is Yesterday A Taste of Armageddon Space Seed The Devil in the Dark Errand of Mercy The City on the Edge of Forever

Season 2:

Amok Time The Doomsday Machine The Changeling Mirror, Mirror The Trouble with Tribbles Journey to Babel A Private Little War A Piece of the Action The Ultimate Computer Assignment: Earth

Season 3:

Spectre of the Gun The Empath The Tholian Web Day of the Dove Let That Be Your Last Battlefield The Cloud Minders Requiem for Methuselah All Our Yesterdays

TAS:

Yesteryear The Infinite Vulcan The Magicks of Megas-Tu Once Upon a Planet How Sharper Than a Serpent’s Tooth

I can elaborate on why these episodes are chosen but I’m afraid my comment would get too long.

Here's five of my favorites, especially because most of them provide some character lore/development that enhances watching the movies:

The Corbomite Maneuver - Kirk's captaincy style
Amok Time - Spock lore
Journey to Babel - fun, ethical dilemma episode, introduces Sarek
Let That Be Your Last Battlefield - moral lesson episode, introduces the self-destruct, and I've found that modern viewers are fascinated by the fact that racism has become alien and confusing to future humans
The Galileo Seven - character drama, features Spock making survival choices that weigh the needs of the many against the needs of the few

I’m a big fan of The Doomsday Machine and The Tholian Web.

Just watch them all!😊👍

All great recommendations so far. I'd add

• TOS: Shore Leave
• TOS: The Trouble with Tribbles
• TOS: Assignment: Earth (almost required context for PIC Season 2)
• TOS: By Any Other Name
• TOS: Cat's Paw
• TAS: One of our Planets is Missing
• TAS: The Infinite Vulcan
• TAS: The Practical Joker
• TAS: The Lorelei Signal

Also all episodes with Harry Mudd. :)

I love TAS and believe all of its episodes are worth watching at least once.

Yesteryear is iconic and essential Spock and Guardian of Forever context but there’s a lot to take in.

TAS has some episodes that may not be the best, but definitely provide many firsts, and as such are an important bridge between TOS and TNG other later extensions of the franchise. So, here’s my list of ‘you really should see them’ episodes and why.

• • The Lorelei Syndrome puts Uhura in command, undoing the damage of TOS Turnabout Intruder ;
• • More Trouble, More Tribbles is a ‘must see’ part of the Tribbles trilogy with the TOS and DS9 a episodes;
• • The Time Trap - lays a foundation for the pocket universe trope;
• • The Slaver Weapon - it’s by Larry Niven and brings his Kzinti into the franchise, which only Starfleet Battles game upheld until they reappear in Lower Decks;
• • The Practical Joker introduces the Rec Room, precursor of the Holodeck, and introduces the ‘space anomaly causes Holodeck malfunction’ trope, with hijinks naturally.

• • The Counterclock Incident - Robert April and time running backwards, again not the best but one not to be missed.

Devil in the dark. City on the edge of tomorrow.

S1e16 - arena is worth a viewing if you have the time.

I'd also add The Enterprise Incident in session 3