My not-particulary-organized observations on rewatching the whole season. I don't quite have the energy to edit this into readable long-form prose, so here are some bulleted lists for you!
Passage of time
I particularly paid attention to any mention of how much time had passed between episodes, or even individual scenes. This show is very tightly edited to keep the action humming along (or sometimes to hide some plot development from the audience so as to have a surprise reveal later), so it's not unusual for a simple scene-to-scene cut to disguise the passage of some significant amount of time.
The most important thing here, which countless people have discovered over decades, is that stardates are totally meaningless. Burnham's logs mention stardate 1207.3 and 1308.9; Harry Mudd (between the two Burnham logs) mentions stardates 2136.8 and 2137.2; and Admiral Cornwell mentions stardates 4789.6 and 4851.5 during her rundown of the disastrous Klingon war during Discovery's absence. Meanwhile TOS uses stardates back in the 1300s, so I think we can agree that this is pretty useless.
Specific episode mentions…
Context is for Kings: while Burnham is doing her "code reconciliation" to prove her worth to Stamets, there is a clock visible on her holographic screen. It starts at 1934 and ends up at 0815, so that passing of a few seconds on screen is over 12 hours of real time. Of course, this is explicitly a "time-passing montage" so that's not surprising, but it gives a hint of how free the show is with surprising leaps of time.
Choose Your Pain takes place about three weeks after the previous episode. Someone explicitly says "these last three weeks" (since the saving of Corvan 2), and there's another mention of "seven months" since the battle of the binaries, which also fits with the previous mention of six months in Context. This episode also places the Buran's destruction at "six months ago", i.e. about a month into the war.
Into the Forest: a good example of "action compression." Burnham indicates their destination is "600 meters this way" and then in the next cut they're there. Moving through hostile territory would have taken a significant amount of time.
Mirror universe arc: each episode seems to cover a day or two. Burnham's voiceover says "it's been two days" since arriving on the ISS Shenzhou in the previous episode. Similarly, in What's Past is Prologue, Burnham talks to Lorca about "what you said on the shuttle yesterday" in the previous episode.
Obviously with the big character revelations, there was bound to be some good foreshadowing early on.
During the first Burnham/Georgiou scene, in the desert, Burnham asks Georgiou, "What would you do if you were trapped here for the next 89 years?" and Georgiou's answer? "Easy, I'd escape." Hints of her mirror-self's badassery.
Voq chokes L'Rell when he thinks she has betrayed him. Then of course, as Ash, he chokes Burnham later. (He grabbed someone else's throat sometime during the season too, but I can't remember who. Maybe Mudd in prison?) Consistent! Proposed Mafia name: Voq the Choke.
L'Rell says to Voq, "I do not want the mantle of leadership. Standing behind you, I am free to move." But of course at the end of the season, she is the leader and now Vash stands behind.
In Butcher they explicitly mention eating Georgiou, which of course L'Rell mentions again to Emperor Georgiou.
Obviously the eye sensitivity thing
Saru: "Captain Lorca is a man who does not fear the things normal people fear."
Lorca's grilling of Tyler about Issaquah is reminiscent of Burnham grilling Emperor Georgiou about Malaysia at the end of the season.
When Lorca and Cornwell are "catching up" there are a few clues that Lorca is not fully familiar with the past. Of Burnham, he says "her story is well-documented" a bit defensively, reflecting the studying-up he has done. And while Cornwell reminisces about old times, he hesitates and she says "You don't remember?" Of course he doesn't, but he manages to deflect. Then of course there are his scars and his sleeping with a phaser under his pillow. (Incidentally I would like to know what those scars—a triangle and some other simple shape, maybe a line?—represent.)
Just before the jump to the mirror universe, Lorca says, "Let's go home."
Cool thematic moments
T'Kuvma's skin is jet black, and of course Voq is completely white. I don't know if that means anything, but it was striking as the ultra-closeup of T'Kuvma's face was the first shot of the whole show.
When Discovery jumps to the mirror universe, it's not the normal "spin and dive" effect. Instead, the ship splits in two and kind of flies apart, instead of just going "down" like it usually does.
Opening titles: do they mean anything? In most Star Trek series, it's just focused on the ship. Enterprise expands that with the theme of exploration and connecting our present time with that of the show. But I'm not sure what I'm supposed to make of Discovery's mixed themes of design schematics, biological things, and dust-to-dust imagery.
Related: Does the name of the episode ever actually appear on screen? I don't think so.
There were a lot of Tilly snoring jokes in the beginning. It's a cheap laugh but I thought they did it pretty well. Could be fun if that continues to be a running gag throughout the show.
LOVE the "Previously on Discovery" announcement, in Klingon, by L'Rell in the last episode.
Burnham says "An isic for your thoughts" is an old Vulcan saying. I wonder if it's related to "Only Nixon could go to China."
Obvious missed opportunity for someone to say "Thar be whales here" after beaming aboard the gormagander.
The ISS Shenzhou has mirrored "rugs" on the floor throughout the ship!
Worst episode rant! Pahvo. First, the blue dust looks exactly like the spores, to the point that I kept assuming they would be revealed to be related, even into the next episode. Second, the silliness of a big crystal antenna on a random planet being able to see ships anywhere in the quadrant in real time just compounds the previous physical silliness of the spore network. Third, I join BionicDave in hating the conversion of Saru into a murderous brute, and then having no consequences come of it. Fourth, of course, who built the tent‽
Saru's eyes: At the beginning of the season they are intensely green, but at the end they are more of a delicate blue. It's probably my imagination, but it seems like they shifted after his "communication" with the Pahvans. (Or, it could just be that the lighting on Discovery seems to be very heavy on the blues, and more broad-spectrum on the Shenzhou.) Things that make you go hmmm.