This is really a matter of taste.
If you're interested in understanding the history of Star Trek, the way Star Trek interacted with American culture and the times it was made in -- watch it all in original broadcast order. This can be looked up on the Internet.
Broadcast order is rather complicated.
- The Original Series has a production order which is different from the broadcast order. It makes more internal sense, but gives less of a representation of how the "culture of the 60s" received the episodes. So
- The original pilot for The Original Series, "The Cage", wasn't broadcast until the 1980s. It's actually quite good. You might want to watch it first.
- DS9 aired simultaneously with TNG and Voyager.
- Star Trek V, Star Trek VI, and the TNG movies aired simultaneously with various series.
- The widely available versions of Star Trek: The Motion Picture and Star Trek VI are NOT the original versions shown in theaters. If you're looking for the true "how the culture received it at the time" version, you would have to dig up the original versions of both of them, and in the case of Star Trek VI, there are no such copies anywhere on the Internet and it was never released on video. In the case of The Motion Picture, most people think the director's cut is better, but it's not 100% period-authentic... the original version was released on videotape and can be found on the Internet if you research it.
- There is a widely-available version of The Original Series with doctored-up modern special effects. Just avoid these and find the as-broadcast copies. The special effects are the least of the problems with The Original Series; the sexism, bigotry, imperialism, and casual genocide in The Original Series are actually easier to watch when the show's visuals are reminding you how dated it is.
If you're not as much of a historian and don't mind some "revisionism" in your Star Trek, you may want to try a simplified order:
- The Cage
- The Original Series (production order)
- The Animated Series (most people would skip this; it's got decent plots but very flat animation)
- Star Trek: The Motion Picture (director's cut)
- Star Trek II, III, and IV in that order (each starts where the last one ends)
- Star Trek V (most people would skip this because it's generally considered bad; it was actually made and released after TNG was running)
- Star Trek VI (this was actually made and released after TNG was running, and is really a sort of "prequel" to TNG explaining why The Klingon Empire Is Different in TNG than it was in TOS and the first four movies)
- Star Trek: The Next Generation
- The various "Next Gen movies", from Generations to Nemesis. These were actually overlapped with the broadcast of DS9, Voyager, and Enterprise, but it doesn't matter that much; there are a few references but they're vague and mostly irrelevant.
- Star Trek: Deep Space Nine. The first two seasons overlapped the last two seasons of Next Gen, and there is some subtle plot overlap, but it's not essential to interlace the episodes. The last five seasons overlapped the first five seasons of Voyager, but there is almost no overlap (there's one "easter egg" guest appearance from a Voyager character on DS9, but you can watch it without having watched Voyager).
- Star Trek: Voyager. Most people would skip this entirely, as it suffered from a lot of really terrible episodes -- especially from heavy use of the "reset button" where huge, traumatic things happened and it's all magically back to normal by the end of the episode. Another name for the "reset button" is "Voyager Syndrome".
- Enterprise.
- Abrams movies. If you liked the rest of Star Trek, you'll probably hate these -- skip them.
Now the thing is, trying to watch all of Star Trek is a pretty big project, the style varies wildly from series to series, and I don't believe you're going to be able to run through it all in one go. You will get bogged down in an entire season of episodes you don't like, sooner or later. When this happens, here's my recommendation.
Every series except DS9 and season 3 of Enterprise can be watched out of order. So can all the movies.
The Original Series: it can be groan-inducingly sexist. Kirk's character is written wildly inconsistently from episode to episode. The "Prime Directive" is introduced, then completely ignored as Kirk works to kill the last member of an endangered species (at least three times), overthrow the entire structure of a civilization (at least FIVE times), etc. When you get bogged down, ask someone for a list of the good episodes ("Devil in the Dark" is one which everyone loves, as is "The Trouble with Tribbles") and the important episodes (ones which later series refer to, like "Space Seed", "The Naked Time", "Amok Time", etc.) and skip episodes such as "Spock's Brain" and "Metamorphosis" which later shows tried their best to ignore entirely.
The Animated Series: no later series refers to it; if you get bogged down, skip to the movies.
The Original Series movies: They're short, just watch them all, but if you're impatient, watch II, III, IV, and VI. Nothing else ever refers to V, and I don't think anything refers to The Motion Picture.
TNG: you are unlikely to get bogged down -- this is the series with the most broad-based appeal -- but if you do get bogged down, first try jumping to the next season, then try skipping straight to DS9. In the unlikely event that you don't like the episodes of TNG, you won't like the TNG movies, they're mostly like average-quality episodes, and they have no important continuity.
Later seasons of TNG set up some stuff for both DS9 and Voyager, mostly related to Cardassians, the Maquis, and Bajor (some related to the Klingons), but DS9 explains most of it again, and Voyager explains most of what it needs. There are some DS9 references to TNG, and one to Voyager, but most of them are Easter Eggs, not essential.
DS9: if you're getting bogged down in season 1 or 2, keep going until you're partway through season 3. If you're really stuck, skip to the last episode of season 1, and if you get stuck again, skip to the last episode of season 2. DS9 takes a while to really get going, and I know a lot of people who were bored by seasons 1 or 2 and completely hooked by the middle of season 3.
DS9 is the most serialized of all the series, and really must be watched in order -- you get really serious payoffs for it. The series ends with a TEN-PART story, eight episodes ending with "to be continued" followed by a two-hour finale.
Voyager: Voyager is off on the other side of the galaxy from DS9, so there are very few references between the two. You will probably get bogged down in Voyager since so many episodes are so terrible. No subsequent series refers to anything which happens in Voyager, and even the TNG movies have only "easter egg" levels of references. If you get stuck, skip to Enterprise.
Enterprise: Enterprise is a prequel. Enterprise seasons 1 and 2 are really really slow-paced and mellow. I like them and I think they're worth pushing through. Season 3 is a large plot arc which got pretty bad reviews. If you get stuck in seasons 1,2, or 3, skip to season 4, and specifically skip to "Home" ("Storm Front" is resolving something from season 3).
Season 4 of Enterprise consists of several very thoughtful and intelligent multi-part stories which are rooted deeply in the 40-year history of Star Trek. If after watching lots of other Star Trek, you get stuck in season 4 of Enterprise, you are not a Star Trek fan. :-) Do not try to watch it before watching lots of other Star Trek.
Now, that said, this isn't the order I originally saw any of this in. My first Star Trek was Star Trek IV: The Voyage Home, which I saw on original release in the theaters (very much a movie of its time, which was 1986), which I love dearly, and I have been perpetually disappointed by how different every other episode of every Star Trek is from it. But I still love DS9, and season 4 of Enterprise, and most of seasons 1 & 2 of Enterprise and most of TNG and the good episodes of TOS and TAS.