Okay, so personally I'm of the school that you should watch in rough order, and some of the callbacks will bear that, as well. So if you are of the same perspective, then of course TOS, first 6 movies, TNG, DS9, next 4 movies, Voyager, and finally Lower Decks. (I skipped some, see below)

You may not be into all of them because all of the series are quite different. Here are my takes on them:

TOS: This one is very slow-paced and philosophical. There are some (cheesy) action scenes and love stories and what have you, but the overarching themes are of discovery and wonder and how, morally, to interact with alien species.

First 6 movies: These are the movies surrounding the original enterprise crew helmed by Captain Kirk and they run from I think 1979 to 1991? Anyway, these have a very different tone and each is their own beast. It's hard to lump them all into one bucket, but they have alot more action, they tend to be a bit darker in tone, and the stories don't really revolve around discovery anymore. It's more military-style. But the ideals of the federation are still there, and they flesh out the motivations of the characters much more than the series ever does.

TNG: This goes back to the roots of the story with lots of exploration of new worlds and alien races. The space quadrant is (mostly) at peace, but the enterprise crew finds themselves often in situations where they are having to battle some creature or race, or some equipment is malfunctioning that could ruin the crew. Also, Q is there, which is always a fun time. This one is much more action-packed than the original series, but still focuses more on discovery. And there are some interesting stories revolving around the enterprise and whether they have any business performing military functions, when they are clearly a science vessel. And I personally find this crew to be the most likeable of the live action series.

DS9: This take place mostly on a stationary space station in the aftermath of an interplanetary war. The federation has taken over the space station to act as a neutral party and ease the transition of peace between the two previously warring factions. And then a wormhole opens up that allows travel between two different quadrants, and this leads to alot of the stories in the back half of the series. I go back and forth on whether I like this or TNG better, but this one has by far some of the best storytelling of the entire franchise. There's a lot of moral ambiguity going on with some of the decisions that get made on this series and it goes to some very dark places. Some of the characters who seem like clean-cut boy scouts will make some decisions that will absolutely change them forever. The latter half of the series takes place during a full-on war and there's a lot of fallout from that. I also encourage you to watch TNG before you watch this because a certain story arc explains the back story of some of the characters.

The next 4 movies revolve around the enterprise D/E crew helmed by Picard. They ran from I think 1994 to 2002. The first one actually transitions from TOS to TNG, and then it's all TNG after that. They vary in quality. The last two, while still good (and one features a young Tom Hardy), are definitely a little cheesier and seem a bit like an extra long TNG episode rather than a theatrical film.

Voyager...It's been a while since I even watched this. I enjoyed it enough at the time because it was Star Trek, but I always found this one to be lacking a bit. I feel like it went a bit into fanfic territory and it was just a little cheesy for me. This one revolves around the Voyager crew getting trapped in a far-off quadrant and their long journey to get back home.

Lower Decks: This one is waaaaaay different than the others. This is a comedic animated series that revolves around a handful of ensigns who live in the Lower Decks of their ship the Cerritos. Two of them are slackers and the others are driven but just too goofy to get ahead. Hilarity ensues. But there are some serious stories too. This one is alot of fun and I highly recommend it, but you won't get alot of the jokes unless you've watched TNG and DS9 first, at least.

The ones I left off:

Star Trek Discovery - I watched the first couple of episodes when it premiered. I did not care for this. It was incredibly cheesy, the characters were forgetful, and it was too much of an action story.

Picard - I have only watched the first couple of episodes of this. I really liked them, but I haven't seen enough of the series to recommend it yet. It's on my list.

Strange New Worlds - This one is also on my list. I have not seen a single episode, though I've heard it's very good.

Enterprise - I've only seen a couple of episodes of this, but what I did see felt very "k-mart version of Star Trek."

I might have missed some stuff, but that's about all I've got anyway.