Out of the characters Burroughs created during his career Tarzan and John Carter are probably the most well known, but the Barsoom series, unlike Tarzan, has managed to fade more into obscurity, despite Disney tried to bring John Carter to modern audiences with a movie adaptation of Princess of Mars. It's not really difficult to see why that has happened though, as a lot of the sensibilities of the series won't sit that well in the modern world. One of the biggest offenders here probably is the female image of the series.
During the 11 books, the part of the protagonist is given to the children of Thoris and Carter on a couple of occasions, as well as unrelated brave men of Mars. What is notable in the series is, that women are always the main drivers of the male characters. For the women, the men are willing to go through hell, but that also defines the role of women as: they are damsels in distress and their white knight has to rescue them. In fact, being a woman in the Barsoom series means, that evil man will capture you and will try to forcefully marry and rape you because that's how it's done on Mars.
Out of these 11 books, only one story has a female character that isn't there just to be rescued, not all the time at least. Tavia from a Fighting Man of Mars is a competent fighter in her own right and overall, at least what comes to the Barsoom series, a strong female character. Unlike the astonishingly gorgeous Dejah Thoris or her daughters whose only role is to be rescued, Tavia is a former slave girl, who knows how to take care of herself. Again, as far it comes to the Barsoom series. Burroughs writes a lot how the women of Mars are feisty and can handle weapons, but it's a rare occasion that women actually do take on arms and in that, almost all the women in the series do tend to feel more like precious objects than actual people.
And then there's slavery. Sure, Barsoom, as it's described in the books, is a different kind of world. And different worlds have different approaches on things, even on those that we see differently. But what's really mesmerizing about the views on slavery is, that it's okay. Not only it's fine with the free population of Mars, it's fully fine with the slaves themselves as well, as a lot of them are described as being pretty happy about it and even very content about their lot as well, as it seems to be deeply ingrained in the society at large.
Then again, there's the hypocrisy of our slavery versus theirs, where the slaves outside city of Helium are often described as unhappy lot, as they've been forced into slavery by an evil, immoral rulers, who also oppress their free people, whereas in Helium it's just a servant job, of which no one pays you and you can be sold. in many stories, there's a lot of talk about freedom, but in midst of all that, even the good guys are pretty happy about keeping slaves, so the freedom isn't all-inclusive.
Sociological issues aside, one complaint also is, that the stories are very formulaic in nature. All the stories really do revolve around a brave warrior going somewhere, where he needs to rescue a woman of his dreams. This all is overly melodramatic, as all the men in the universe seem to fall in love immediately, after which they're willing to move heaven and earth in order to secure that love. And all the women, of course, instantly responded to that love, despite neither party really knows each other or have just spoken to each other briefly.
This melodramatic love also takes the hapless lovers into adventures, which consists of them constantly finding a long forgotten civilization of Barsoom, getting caught by them and finding a way out. And more often than not, it all ends happily, when the great navy of Helium rides to the rescue with John Carter at the helm. This all ends up so structurally sound, that it's relatively safe to skip chapters of text without a feeling that you'd lost something important.
Burroughs is a fluent writer and that makes the Barsoom books very easy to read. The overly familiar structure they all have does it make feel that his heart really never was in it fully, at least what comes to the latter half of the series. He uses, increasingly more, a lot of stock description, where he describes things, like Martian measurements or technology, book after book.
Of course, a lot of the structure in the stories is caused by the nature of how they were originally published. The Barsoom series originally was a magazine serial, so each segment just had to be adventurous and full of action, as that made the people get the next issue as well. This structure, however, makes the book form feel at times a bit parodic, intentionally or unintentionally, because it's just so overstuffed with action, especially in the first three books, where Carter keeps killing more and more enemies by the chapter so, that he's pretty much a superhero in the end.
Carter's superhero-like stature probably also explains why he was put on the sidelines after the first three books. Burroughs was most likely feeling a bit tired of his own hero, who just was too good for his own good: no matter what happened, he always rose to the occasion. He just wasn't designed to lose. But then again, none of the other heroes was designed to lose either, but they weren't always as fun to follow on their path to victory. Thuvia, Maid of Mars is a good example of this, as its male protagonist Carthoris feels like a lite and bland version of his father Carter and Thuvia, the more interesting of the two characters are left in the background to be the damsel in distress, when the story itself could have been much better had Burroughs let Thuvia been even more active part in the story.
If you're interested in reading the Barsoom series, I'd recommend reading the first 3 books in order, as those give you a good, solid foundation to proceed upon. The rest of the books, at least up to the 10th, can be read in any order basically, as they all are stand-alone stories. Of course, some of them refer to things happened in the previous books, but there's nothing really that drastic about them, that would require reading them in order, especially when they don't star John Carter or his kin.
Llana of Gathol can be read pretty much at any point and it can be read as a separate entity, as it is pretty much a parody of the series. It's fun and Burroughs used the basic structure of the other stories to bring up silly, coincidental nature of many of the things that happen during a typical Barsoom story. The last short story collection, John Carter of Mars, can easily be left unread, as neither of the stories is that great and the better of the two stories are unfinished.
As copyright for many of the Barsoom stories has elapsed, even worldwide, so it is pretty easy to find an eBook version of many of them. Project Gutenberg for an example is a good source for free editions. If you prefer a store, the Complete Works of E.R. Burroughs that is available at least in Google Play store contains 70 stories written by the man. Similar collections are available through Amazon as well. And libraries are a good bet as well, especially if you prefer a print book.
My take on the individual books:
A Princess of Mars
The Gods of Mars and The Warlord of Mars
Thuvia, Maid of Mars
The Chessmen of Mars
The Mastermind of Mars
A Fighting Man of Mars
Swords of Mars
Synthetic Men of Mars
Llana of Gathol
John Carter of Mars