There are some complicating factors that can lead to different (or streamlined) sequences though. First, most of the books center around a character named Miles Vorkosigan, but the first two are told from the perspective of Miles' mother, Cordelia Naismith. Second, the initial installments of the series were written and published out of chronological order, so there are plenty of folks out there who've read the series that way. Third, Borders of freakin' Infinity (more on this below, but it's a collection of novellas that can confuse the order). Fourth, most of the books have been collected together in omnibus editions, which complicates things a bit, but if you want to read the series chronologically, they're actually pretty well organized. Fifth, and the reason I struggle with the whole reading order thing, is that the height of the series starts about 8 or 9 books in... This is not to say that those first 8 books aren't good, just that the series got better than I ever expected around that time with an amazing four book run starting with Mirror Dance and concluding with A Civil Campaign.
Every book in the series tells a discrete story. There are no cliff-hangers, but there are a bunch of character-centric multi-book arcs. Interestingly, the series could be read almost as a series of pairs, and the omnibus editions are mostly built around that notion, with the novellas from Borders of Infinity thrown in for good measure. Aside from attempting to read the second of a pair first, I suspect you could try to get into the series almost anywhere along the way. Before I go further, it might be useful to list out the series, publication dates, and omnibus editions:
|Falling Free||1988||N/A||Independent novel set 200 years before main series.|
|Shards of Honor||1986||Cordelia's Honor||Told from Cordelia's perspective.
Kaedrin Reviews: Shards of Honor, Barrayar
|The Warrior's Apprentice||1986||Young Miles||Kaedrin Reviews: The Warrior's Apprentice, Mountains of Morning, and The Vor Game|
|"Mountains of Morning" from Borders of Infinity||1989|
|The Vor Game||1990|
|Cetaganda||1995||Miles, Mystery & Mayhem||Ethan of Athos is an independent story.
|Ethan of Athos||1986|
|"Labyrinth" from Borders of Infinity||1989|
|"Borders of Infinity" from Borders of Infinity||1989||Miles Errant||Kaedrin Reviews: Borders of Infinity, Brothers in Arms, Mirror Dance|
|Brothers in Arms||1989|
|Komarr||1998||Miles in Love||Kaedrin Reviews|
|A Civil Campaign||1999|
|Diplomatic Immunity||2002||N/A||Kaedrin Review|
|Captain Vorpatril's Alliance||2012||N/A||Told from Ivan's perspective.
So the first thing to note is that Falling Free, while technically part of the series, is an extreme prequel and doesn't really involve any of the characters (it's set 200 years before the rest of the series). As such, it's almost completely independent of the rest of the series. I say "almost" because I've heard it would be good to catch up with this one before Diplomatic Immunity. Also, I neglected to mention a short story called "Dreamweaver's Dilemma" which is apparently also an extreme prequel that's not closely coupled with the rest of the series. Of course, I haven't read either of these, so I can't say for sure (Update: I've since read both, and they are indeed not closely coupled with the rest of the series, I would skip them and come back later). I do plan to catch up with them at some point, but the real meat of the Vorkosigan Saga starts with Shards of Honor.
Shards of Honor is where I started the series, though it appears that many people bypass Shards of Honor and Barrayar, and start directly with The Warrior's Apprentice, which is when Miles first shows up (well, there is a short scene in Barrayar where you see him as a young child, but that's from his mother's perspective...) There are pros and cons to each approach. Starting with Shards of Honor and Barrayar gives you a lot of background on the universe and characters, while The Warrior's Apprentice will get you into the series quicker. Personally, I opted to start with the Cordelia books. I'm something of a completist, but it worked really well for me. The other option is to read the books in order of publication, which will have you ping-ponging from Cordelia stories to Miles stories and back again a few times, as well as being all over the internal chronology... but I'm sure it would work too.
The most confusing thing I encountered in the series, though, is Borders of Infinity. This is a collection of three novellas (including one called "Borders of Infinity", just for added confusion), which in a lot of other arenas, means that you can probably skip them... but I would strongly advise against that, actually. "Mountains of Mourning" is quite possibly the best story in the entire series. "Borders of Infinity" is a really clever prison story, and the events in that story - some of which rubbed me the wrong way at the time - pay off huge in Komarr (I have no idea if that was always Bujold's intention, or if she just thought of it later, but it was a fantastic revelation in any case). "Labyrinth" is the most unusual of the bunch, but it also introduces one of my favorite side characters from the series, Taura. Now, these stories were originally published as part of one collection, but the three stories all take place at varying points of the chronology. The omnibus editions do an admirable job mixing the novellas into the series though, which lessens the confusion quite a bit. The only thing lost, then, is the narrative glue between the stories, but that's only about 5 pages or so (even still, it takes place between Brothers in Arms and Mirror Dance, making it a difficult thing to fit in - you won't really miss it). Anyways, there are a bunch of options for Borders of Infinity... it might even make an interesting introduction to the series, though it's always hard for me to judge (I'd still recommend starting with Shards or Warrior's).
Ethan of Athos is another book that is noteworthy for its independence from the rest of the series. Indeed, Miles is barely even mentioned, though one of the series' recurring characters, Elli Quinn, plays a prominent role. It's an interesting story, probably one of the least mainstream of the entire series, but it's also very independent. There are some small references to it in later stories, but nothing big enough to say this needs to be read in order (though, completist as I am, I did). If you're looking to get to the amazing four book run starting at Mirror Dance, you can probably skip this one.
So I think that covers all of the exceptions and divisive parts of the series. There are a lot of books that pair together well, and I think the omnibus editions do an excellent job latching them together. Incidentally, just because something isn't part of an omnibus doesn't mean it's not worthwhile. I think my second favorite story in the series is Memory. Also, just because something is a novella or a short story doesn't mean that it's not worth reading. I've already mentioned that, but it bears repeating. Even "Winterfair Gifts" was a great story (which, I believe, is only really available as part of the omnibus).
This series has probably been my favorite recent discovery. It's a tricky thing, and I think there's an interesting discussion to be had about series like this. I have to wonder how good something like Memory or A Civil Campaign would appear to an outsider who didn't have so much background on the characters or the universe. It certainly worked wonders for me, but it's hard to express that because in order for anyone else to get that feeling, they have to read several books into the series to get there... Tricky. But that's a discussion for another day.
Update: Added Captain Vorpatril's Alliance, some additional notes.