I'd probably just start the PIC era in 2399. We only see the 2380's in a handful of flashbacks. Prodigy definitely has some PIC connections, but it's far more a continuation of Voyager, and it starts like a year after season 5 of Lower Decks.
Grain of salt since I can’t remember where I heard this, but my understanding is that TOS-ENT and the first ten movies is generally a single combined license, but all of the new shows need to be licensed individually, I’m not sure if the Abrams films are licensed as a package or individually, but they’re separate from the other films.
Fun fact: The TNG episode of the same name was originally intended to star Patrick McGoohan as Ira Graves.
Not only do you not need to catch up with Disco, but the end of season 2 of Disco doesn't dovetail to the beginning of Strange New Worlds super-well when you watch them back-to-back; if anything, they feel like they lightly contradict each other.
Though if I were to recommend a single episode it would be "Through the Valley of Shadows" from season 2, which is probably the only "important" episode as far as Strange New Worlds goes, though the key stuff gets recapped in SNW.
The Short Treks episode Q&A gets some references in the series too, though if you don't watch it you likely won't realize you missed anything.
This is the first time we've learned about the making of bloodwine, and seemingly the first confirmation that it's made from actual blood of some sort of worms (or worm-like plants?) grown in the ground.
The band near the beginning features an accordionist playing the same style of concertina Ron Taylor's Klingon chef played in DS9: "Playing God".
K'Ellara's "boob window" design was featured on Lursa and B'Etor starting in TNG: "Redemption".
They mention Martok fighting in a bar in the Ketha lowlands. Martok said this was where his family was from in DS9: "Once More Unto the Breach". We later saw the Ketha Province in Star Trek Into Darkness.
"Experience bij!" is a reference to the Star Trek: The Next Generation Interactive VCR Board Game. Whenever Robert O'Reilly's (non-Gowron) Klingon said a player would "experience bij" this meant they had to draw a card, usually resulting in something bad happening to them. https://www.youtube.com/watch?v=VjAvGNn20Y8
As mentioned above, Fek'lhr was a Klingon "devil" seen in TNG: "Devil's Due". There are legends of Klingons killing their gods (DS9: "Homefront"), so it's possible Kahless killing Fek'lhr is in reference to that.
Councilor Enaj, named only in the subtitles, is Jane spelled backwards.
I guess I'm saying the obvious here, but Boimler's transfer was to the Titan starting in LD: "No Small Parts" and lasting through LD: "Kayshon, His Eyes Open".
Consistent with earlier episodes of Lower Decks, Klingon blood is pink as it was in Star Trek VI and I believe some of Discovery. Otherwise Klingon blood is generally red.
"I can see Kahless! The first one! The original one!" Presumably in contrast with the clone from TNG: "Rightful Heir".
Mary Chieffo is credited in this episode. She played L'Rell on Star Trek: Discovery. I'm not great at recognizing her voice, but she presumably played K'Ellara or Enaj, though maybe I'm just typecasting her and she played food critic Madame Gonald. Screen Rant says she was K'Ellara. https://screenrant.com/star-trek-lower-decks-mary-chieffo-discovery-klingon-comeback/
In any case, one of the female Klingons was played by Dulcé Sloan of The Daily Show. She teased this in an interview last February. https://people.com/dulce-sloan-on-new-book-exclusive-8557416
EDIT: https://trekmovie.com/2024/11/07/recap-review-star-trek-lower-decks-serves-up-a-delicious-cold-dish-in-a-farewell-to-farms/ also says that Chieffo was K'Ellara. It also says that Sloan was Gonald. Since Sloan said she played a Klingon, I assume she was playing Enaj as well.