as I am winding down my TNG thon - i finally can list the episodes that I feel are important to watch.
1: Encounter at Farpoint
where we meet our plucky crew, and realise they are all very very cranky. also the last time we see Troi in proper uniform until Season 6. Also. Q. Hi Q.
2: The Neutral Zone + Q, Who + Best of Both Worlds + Family.
we first meet Romulans, but they aren't the ones you are worried about, it's the thing that made the big hole in the planet. Then we see Q again (plus the awesome Q v. Guinan fight) actually introducing us to the creepy aliens the Borg, then to get to the Best of Both Worlds/Family where we end that saga for the most part, as the other Borg episodes in the series tends to be another arc and not really linked to this one imo.
3: The Measure of a Man + The Offspring
this is hands down one of my favourite episodes. I don't even think you need to watch any TNG before this episode (truly) and you still wouldn't be "lost" it all really makes sense. one of the best "Data yearns to be human/has rights/but is seen as property" episodes - which ties into the "sequel" of sorts with The Offspring, as Data's child is seen as property, and the lengths Starfleet tends to wrap itself into justify being horrible, ultimately Lal being shut down due to her emotional overload.
4: The Emissary + Sins of the Father + Reunion (Redemption if you squint)
Worf finds his first wife so to speak (more of a mate than what Jadzia ended up being for Worf), but she turns him down, then we learn a lot about Khitomer and how much Romulans took from Worf, and that Worf's dad is a traitor - but no, everyone knows it wasn't Mogh it was Duras, and this really "Honour is everything" species is willing to mud sling a dead man to keep the peace, following into the Reunion where we now have K'hlyar wanting to truly mate with Worf, and a Son, but Worf can't do anything because he's shunned, leading to her death, Worf raising his son and a really big "I give zero craps being a starfleet officer). (plus Worf's first Formal Reprimand. Which adding to the second one he got in DS9, explains a lot of why he chose his future career path)
5: Yesterday's Enterprise + Sarek + Unification
in where we see a really good Historical episode, give Tasha's death more meaning (though it gave us more Worf, so thanks Denise for wanting to quit), leading up to seeing Sarek (one real last time) and actually giving a voice to the emotions we always wondered, into having a great look/insight how Romulans do things (like why try to always annex Vulcan?) what happened to Tasha, and a bridge between Sarek, Picard and Spock that is really well written. quasi arc
6: The Perfect Mate.
there are a few "Picard finds someone" episodes, but I love this one. I think you ultimately get this sense that Picard fell for this woman, you feel heartbroken at the end of it, and then the humour was on point.
6: The Chains of Command
I! SEE! FOUR! LIGHTS! no really, , this is hands down one of the best episodes of the series, I think it really puts you in the frame of mind of how Cardassians really are and how everyone keeps waltzing on this line of "hey you know we have this peace but we really do not like you or trust you" then you have Jellico who tells deanna to dress properly (finally), who kinda hates the hand holding Picard has this crew on, then you get the whole torture scene. so awesome.
7: Lower Decks
I would have added "The First Duty" (in my HM) as something to watch, but I don't think you need it truly to appreciate this episode, and I honestly feel that a great way to have more startrek is to have the lower desk style. written well it won't be too "high-school" feeling. anyway. we have a great intermingling of senior staff, Ensigns fighting for promotions, and sacrifice.
8: Ensign Ro + Journey's End +Preemptive Strike
the first Ensign Ro episode, the last Ensign Ro episode, sandwiched in between a Wesley choosing to screw Starfleet episode. (You could skip Journey's End, I mean it's not a MUST WATCH) but I feel that these three episodes explain the Bajorans (specifically) why they are in Starfleet (makes you wonder who sponsors them to join Starfleet), leads up the TNG portion of the Maquis and how Starfleet is wrong in a way in how they are handling the DMZ, the Treaty etc.
9: The Inner Light + Lessons
we had the argument if Inner Light is overrated or not - but I do feel this is one of the episodes that everyone should watch at least once, and then make the decision on their own. Lessons (where Picard falls in love), is the direct sequel as we hear how it impacted him and how he got into music because of these experiences.
10: Tapestry + All Good Things....
I linked this because, Tapestry needs to be in the top ten. and I didn't know what to kick out So you watch this one. Where we find out Jean Luc was very not diplomatic growing up, and Q'.. just being Q. - followed by Q telling everyone "Hey guess what. everything from Farpoint - still going on!" and we try to unravel the mystery, and we are left wondering if Humanity was truly on trial, or Q always forcing the crew (and ultimately us) into thinking beyond what we know, see etc.
Honourable Mentions:
The Price, A Matter of Perspective, clues, Nth Degree, The First Duty, the Chase. oh and the Drumhead and Force of Nature.