A true fan does not watch a show just once. They watch again and again and again. For some shows, this is straightforward: You (re)start with the pilot and go forward until the finale. For the multi-series, multi-movies, multi-timelines entity that is Star Trek, however, the order in which you should go through the episodes is a rich topic of discussion among us devotees.
If you are a complete newcomer to Star Trek, you should watch everything in the original broadcast order. Even this seemingly straightforward idea is somewhat tricky to implement for Star Trek, because you should know how to intermingle DS9 episodes with those of TNG and VOY, considering that DS9 was broadcast parallelly with those two other shows and some TNG movies, but this wonderful list (which also formed the basis of my list) solves that problem for you. (I'm assuming that you know the acronyms for the various series, as you have just seen.) My aim here is to provide a listing for your second watching, where you will be digging deep into the details.
The correct thing to do in that second watching is to follow the in-universe chronological order. This is also not as simple as it sounds, since one quickly runs into difficult questions about where to fit the alternative universes, the wonderful time loop introduced in the First Contact movie, and just how to watch the finale of ENT, which actually takes place within an episode of TNG!
Well, I have solved those problems for you. Below, you will find a complete list of everything in the Star Trek canon (TV series, movies, the Short Treks (ST) episodes, animated shows, everything!) ordered in the way they should be ordered for optimal viewing in in-universe chronological order.
How did I handle the tricky cases?
The causality loop introduced by the First Contact movie (Should I have given a spoiler alert at the start? Well...) is of course impossible to linearize, since it is a loop. So I take the most sensible way out. A complete newcomer watching in this order will of course be confused by those cyborgs being discovered on Earth in the ENT episode Regeneration, but the writers of that script took the best care they could of newcomers, and I have already stated that this list is not for newcomers.
Something similar applies for the time loop that sort of has to exist between The Squire of Gothos and Wedding Bell Blues.
The places where stand-alone episodes/movies taking place in alternative timelines appear have been determined according to the point at which the events they depict have been caused by events in the prime timeline. For instance, the ENT episodes taking place in the Mirror Universe sit nicely between the ENT episodes in the prime timeline, since the Tholians of the ENT era caused what happens in that story. On the other hand, the Kelvin timeline movies should be watched after the Short Treks episode Children of Mars and before PIC, since they happen because of something Spock did between those two stories in the prime timeline. I think you'll find that this approach makes sense.
The Section 31 movie starts in one time in one universe and then continues in another universe a long time later. I chose to place it where most of the action occurs.
Of course, some episodes get thrown to places very far from their broadcast order. So, for instance, you see a particular episode of VOY at a much, much later time than all the others. That is the fun of this project.
Common sense is used to resolve cases where temporal coordinates are not clear.
Stardates are not useful for this purpose, as SNW and TOS make abundantly clear. Just assume that they represent something more complicated than our usual one-dimensional concept of date, if you wish to find a way to live with that.
The silliness at the very end of The Trouble with Edward is not part of canon and should just be ignored.
Without further ado, here is the giant list in correct viewing order. Please let me (sayster@gmail.com) know if you see any problem.