Nonlinear format is typically not a popular medium for storytelling. The reason for that is because it is generally harder for a lot of people to follow. The Melancholy of Haruhi Suzumiya is a twenty-eight episode anime told in nonlinear format with fourteen episodes in each season. The first season was broadcasted in nonlinear format in 2006 and each episode actually had two previews of the following episodes. One character would show you the next episode for the broadcast order and another would show you the next episode for the chronological order. Consequently, this means that each episode has two different episode numbers (three if you count the second season... which I'll get to later). My advice is to watch it in the broadcast order first. It's just more interesting that way. Yeah, it's confusing for a bit, but it's just plain better to watch it the way it was intended to be watched. So 1-2-3-4-5-6-7-8-9-10-11-12-13-14 is the correct order no matter what anybody says!

However, if you are curious as to what the chronological episode order would look like then I'll list that, too. 2-3-5-10-13-14-4-7-6-8-1-12-11-9 is the chronological order. Be warned that if you were to watch it in chronological order than the show would reach its most dramatic point at episode six (14, in other words) and then everything after that is just fun and games, really. Not quite so interesting. The final episode in chronological order (9) is not really anything special. It doesn't feel like an ending. In broadcast order the dramatic climax is the final episode (14) and we actually have a buildup of suspense throughout the series because there really is no telling what will happen (or has happened, as the case usually is). We get a few brief images here and there in broadcast order but nothing too obvious until the big reveal at the end.
Chronological order gives it to you all up front on a silver platter with little payoff at the end. After you watch the sixth episode (14) there really isn't a point in continuing with the show unless you just want to for kicks. Please do keep that in mind before you jump on the chronological bandwagon. The last real interesting show in terms of actual plot would be the eighth episode (7), but it still wouldn't touch 14. The rest is still good and fun, but it just leaves you with nothing special to look forward to. Not even season two and, like I said, I'll get to that later.
If you have seen the show in its proper order already then feel free to watch it in chronological order now, but don't do it if you haven't ever seen it!
In broadcast order you don't know exactly what you are getting, though. The narrator Kyon talks fast through each episode and sometimes the things he says go over our heads because he is referencing events that we don't know about yet and mentioning people we don't know yet. But that is what makes things interesting in this show.
Nagato Yuki sometimes has glasses and sometimes doesn't. How she loses them (or gets rid of them for all we know) is important in terms of the plot, but we don't know that until we actually watch the episode in which her glasses finally stop being used.
It's the little things that make the broadcast order important. We pay attention more to little details when you are not sure what will happen next. That's just human nature, I think.
Now I want to bring season two into discussion. When season two was aired it was aired while season one was being re-aired in chronological order in 2009 because season two isn't really season two at all. In fact, season two fits into the chronological order of season one like a glove.
So let's say we organize season one into chronological order again. 2-3-5-10-13-14-4-7-6-8-1-12-11-9. Now let's add season two into the mix. I'm going to differentiate the two seasons by adding an "S" in front of the season two episodes so you hopefully won't be too confused. 2-3-5-10-13-14-4-7-6-8-S1-S2-S3-S4-S5-S6-S7-S8-S9-S10-S11-S12-S13-S14-1-12-11-9.
However, I strongly suggest that you don't watch season two until after you've finished the broadcast order of season one. Trust me on this.
Season two is tough to make it through. The first season is really good despite its wacky episode line-up, but season two is grueling because of the dreaded Endless Eight episodes.
Episodes 2-9 of the second season make up the Endless Eight and only the most die hards can make it all the way through. I couldn't. I admit it. I skipped four of the episodes because I just couldn't take it anymore. Why? Well, I ought to tell you about the plot of the show first.
Suzumiya Haruhi is a God. Maybe. Only she doesn't know that she might be God. She's weird, too. Other than the whole "might be God" thing. She wants to meet aliens, ESPers, and people from the future. She doesn't care who knows it.
Kyon (a nickname, his true name is never mentioned) is a laidback and snarky guy. He wants life to be simple and he enjoys keeping a low profile. One day Kyon starts up a conversation with Haruhi. And Kyon suggests that she ought to start a club if she is so bored. Eventually he finds himself a member of her school club "SOS Brigade" and he is forced keep up with Haruhi's eccentric activities or else the whole world might end. Quite literally.
Nagato Yuki (alien), Asahina Mikuru (girl from future), and Koizumi Itsuki (ESPer) soon join and thus the SOS Brigade is formed with Haruhi as the leader and Kyon as the hapless ordinary (maybe) human of the bunch.
Of course, Haruhi doesn't know that she is surrounded by the very people she is trying to seek out and no one seems to want to tell her the truth except for Kyon. And Kyon himself isn't sure if he believes she is God at first.
Now I am going to talk about the Endless Eight from season two.
It's the end of summer. The final two weeks to be exact. Haruhi calls up her club and makes plans for them to enjoy the final days together. The festivities include goldfish scooping, cicada catching, fireworks, attending festivals, and holding part-time jobs. All of this is wrapped up in episode 2 and at the end Haruhi seems to have something else she wants to do, but no one else makes a suggestion. With one day left in their break they go their separate ways and Kyon fails miserably to get his homework done before the next day.
But the next day doesn't come and it will not come for a very long time because Haruhi regretted not doing something. What? Well, that is the million dollar question.
The final two weeks in August repeat exactly 15,532 times. Kyon, Koizumi, Mikuru, and even Haruhi seem to not be aware that time is repeating although all of them except for Haruhi experience some serious deja vu at times.
Nagato Yuki knows exactly what has happened though because she is given the arduous task of "observing" all events related to Haruhi by the order of the "Thought Entity." This certainly explains why Yuki always looks bored as hell no matter what they do over their summer get-together.
But it doesn't explain why the creators of the show made eight episodes almost exactly the same. This is why a lot of people don't like the second season. So much of it just seems wasted on repeating one episode. Yes, each of the Endless Eight episodes are a little bit different. The characters wear different uniforms, sometimes the lines they say are different, sometimes they say the same lines but at different places or different times. It's all a small variation and if that's your thing then you'll fall in love with the Endless Eight.
My advice? Watch episodes 2, 3, and then 9. Or in place of 3 you can substitute 4, 5, 6, 7, or 8 because it really doesn't matter much which one you choose. They are all essentially the same despite little differences. But of these Endless Eight episodes you really only need to watch three of them. Once you finish the second season (and thus the entire show) then feel free to go back and see what you missed of the Endless Eight.
There is a movie that is about three hours in running length that also fits into the series and the movie is called The Disappearance of Haruhi Suzumiya. I probably have to buy that in order to watch it because Crunchyroll doesn't have it, but after watching the series I think it'd be worth it. Endless Eight or no, this show is really good. A movie sounds great. Three hours sounds even better.
I'd love for there to be a season three as long as I don't have to go through an Endless Eight Part II. That really knocks the second season down quite a few notches, but the first season and the rest of the second season are really good.
I recommend this show for its wackiness and its intrigue.
I should also mention that Suzumiya Haruhi is the ultra director of this show and that it is produced by the SOS Brigade. Maybe there is a good reason this show is so bizarre...