I've been re-watching the first season of Person of Interest. My framework for doing this has been: which episodes should people really watch?
I'm somewhat frustrated to discover, the answer is: all of them. Of the 23 episodes, I'd recommend watching at least 20 of them. The other three aren't bad (they're better than a couple that I'd have to recommend for plot reasons), they're not not essential. And even they progress the plot.
The first season of PoI could be a great lesson for TV writers in terms of structuring a season - it's such an evolution over the conventional 'big bad' format. If you take the 'big bad' format: there's a monster (or case, etc) of the week, and then a season-long arc building up a 'big bad' in the background, a higher-level challenge. POI likewise splits time between COTW and arc, but it's arc isn't just one arc: it basically has four major external arcs (the mysterious backgrounds of both lead characters, law enforcement trying to catch them, and two different criminal organisations). Not to mention the 'internal' individual character arcs of the two protagonists, and to a lesser extent the supporting cast, and the relationship arc between the two protagonists. These arcs are woven through the episodes - so no arc has to feel strung out too long with too little material, as a lot of modern TV shows do (some, in fact, feel too hasty) - and, like an MPOV fantasy novel, there's a kind of feeling that you have to watch on in case the next episode follows your favourite arc. Meanwhile while each arc comes to a greater or lesser level of conclusion, several of them are really just being set up for later seasons - so in later seasons the arcs can feel 'important' and weighty even if they don't get many episodes, because they've been going on for years. Indeed, one of the most important arcs at the end of S1 is a fifth external arc that's barely presaged in the season - the finale is more a launching pad for a new plot than a conclusion of an old one (though it is that too). It's not a perfect season, but it's a remarkably good one - I remembered how I liked it more as it developed into less of a procedural show, but actually it's about a good a procedural as you're ever likely to find!
FWIW, my "abridged version" would be all the episodes barring 5, 12 and 16. And I have a really soft spot for 16. None of these are 'don't watch' episodes, they're just 'not essential' episodes. Oh, and you could also maybe skip 14.
[the worst episodes may be 2 and 3 (and the main plot of 1 is weak too), but they're essential for their masses of set-up and backstory].
However, if that abridgement is too long, a 'super-abridgement' would be 1-4, 6-7, 10-11, 13, 15, and 19-23. Skipping 8-9 and 17-18 will cost you some character backstory and minor plot developments - a few things will be unexplained, but it should basically make sense.
And an absolute 'I just want the core plot' indispensable episode list would probably be 1-3, 7, 10-11, 19-23. [you could in an emergency drop 3, 7, 19, and I guess if necessary 20. And i guess 2 if you're smart and paying attention to the others?].
The best episodes, btw? 4 (Cura Te Ipsum), 7 (Witness), 11 (Super), and 23 (Firewall). 20 (Many Happy Returns) is a fan favourite and not without reason, though I might prefer 22 (No Good Deed). So I suppose a really minimal 'i don't care much about continuity but just want goo episodes' list might be 1, 4, 7, 11, 20-23. Which is already as long as some entire seasons of recent shows...
Before that, I rewatchd the first season of Halt and Catch Fire. It's... so much worse than it became. It's still an enjoyable show in the first season, but I can understand why many people didn't give it time - it's much more melodramatic and over-the-top than it would become in its genius later seasons. It's amusing, in a way: a lot of the first season is a debate between doing something well and doing something profitably, and the show sort of illustrates that: the first season tries to do it profitably (and ends up sort of trapped - too 'prestige' for mass market, but too cheesy for critics), but by the fourth season it's given up on any pretence of commercial viability and just sets out to be one of the greatest seasons of TV ever...