Not the easiest or fastest list to compile, but episodes that hold up well and feel closer to TOS in tone and/or spirit than others might include:
TNG
Unnatural Selection (taking "The Deadly Years" in a whole new direction. If the TOS fan likes the progenitor episode, then show them this one.)
Contagion - has a TOS-esque vibe with the portal and Romulans but is still decidedly TNG
The Defector - build and continues on Romulan lore that TNG was right to include despite Roddenberry being against returning foes originally
Where Silence Has Lease - okay, the TOS fan may quickly point out that the big void area is indeed reminiscent of a certain TOS episode (as a bonus round, let them name it

) but once they get beyond that it's a decidedly different situation and is creepy as all heck. Very Twilight Zoneish in some ways
DS9
Paradise (the only other captain that might be as defiant is Kirk)
For The Uniform (that's what Kirk would do if he felt as cornered)
The Way of the Warrior (great Klingon action and you know the Prophets helped too)
The Visitor (As much as I love whining about the big hole, a first-time viewing packs a worthy punch and it's so well-acted that subsequent viewings still buoy it from the plot point.The main plot is good enough to overlook it as well. It's a great Sisko story in its own right.) This one isn't as much one a TOS fan might like out of the blue, but seeing enough DS9 and liking it, it's a perfect one to branch out with as it shows more love in this episode than Kirk ever did in 79 episodes.
VOY
Living Witness (It'd make you wish TOS explored it. )
Blink of an Eye (they nicked the title of the TOS episode, but created a refreshingly innovative concept with the TOS concept as a springboard that still has enough TOS-esque elements and could arguably fit in with Wink, All Our Yesterdays, et al. )
Scorpion (if you threw out "The Visitor" because it's not TOS-enough, this one definitely has a TOS vibe, especially the music that feels like a homage. Sadly, it does require knowledge of the Borg, requiring one to see Q-Who and The Best of Both Worlds and none of the others prior to Scorpion...)
This post is certified Trelane-free (Q-less, hehe...)