Warning: This is long.
A reminder: Percy Jackson and the Olympians deals with the Greek gods. The Heroes of Olympus deals with the Greek and Roman gods. The Kane Chronicles deals with the Egyptian gods. Magnus Chase and the Gods of Asgard deals with the Norse gods. The Trials of Apollo has Greek so far, and might have Roman.
Warning: There may be some slight spoilers included here. I tried to put them in spoiler quotes, so don't click those if you haven't read them yet. There might be a few minor spoilers not marked, but I tried to mark them all.
I will present two orders here, chronological order and the order in which they should be read first.
#Order in which they should be read first
This is mostly in chronological order, but it's ordered more in how they came out and how they will make the most sense. When there are 3-5 books just listed in order, that's the order that the series goes in.
Percy Jackson and the Olympians
The Lighting Thief.
The Sea of Monsters.
The Titan's Curse.
The Battle of the Labyrinth.
The Last Olympian.
This is the original series; therefore, it should be read first.
Now you should read the short story collection The Demigod Files, as most of the stories take place during/after Percy Jackson and the Olympians.
The Heroes of Olympus
The Lost Hero.
The Son of Neptune.
The Mark of Athena.
The House of Hades.
The Blood of Olympus.
This is the sequel series to Percy Jackson and the Olympians, and is a continuation of the stories.
- After this would be a good time to read The Demigod Diaries, a collection of short stories, which have stories from The Heroes of Olympus.
That's the obvious part, as those are all with the same cast of characters. Now..
The Kane Chronicles
The Red Pyramid.
The Throne of Fire.
The Serpent's Shadow.
These are sort-of in the same universe (see the other question) but they are with entirely different characters. The series makes several references to Percy Jackson and the Olympians, though, so read that first.
Now, you should read the crossovers, as they only make sense if you've read both The Kane Chronicles and Percy Jackson and the Olympians/The Heroes of Olympus:
The Son of Sobek.
The Staff of Serapis.
The Crown of Ptolemy.
Now, you should read The Trials of Apollo and Magnus Chase and the Gods of Asgard together, as they take place at the same time and may have spoilers if not read together.
The Trials of Apollo is the third Camp Half-Blood series. Magnus Chase and the Gods of Asgard focuses on Annabeth's cousin, and is sort of a continuation of The Heroes of Olympus, as it makes several references, and Annabeth appears in the books.
This is as in chronological order as I can figure out, based on the books that are out:
- The Trials: The Hidden Oracle.
- Magnus: The Sword of Summer.
- Magnus: The Hammer of Thor.
The rest haven't come out yet.
#Chronological order
The chronological order is great for a re-read, and probably isn't bad if you're on a first read, but the above is better for first ;)
- The Demigod Diaries: The Diary of Luke Castellan.
This is a story set 5 years before the events of The Lightning Thief, focusing on Annabeth Chase, Thalia Grace, and Luke Castellan.
- Percy Jackson and the Olympians: The Lighting Thief.
This is the beginning of the stories, which introduced us to Camp Half-Blood and all our favorite characters.
- Percy Jackson and the Olympians: The Sea of Monsters.
This is set a year after The Lightning Thief, and focuses on Percy & Co's adventure in the Sea of Monsters, aka the Bermuda Triangle.
- Percy Jackson and the Olympians: The Titan's Curse.
This is set half a year after The Sea of Monsters, and focuses on Percy & Co's adventure with the Hunters of Artemis.
- Percy Jackson and the Olympians: The Battle of the Labyrinth.
This is set half a year after The Titan's Curse, and focuses on Percy & Co's adventure in the Labyrinth.
- The Demigod Files: Percy Jackson and the Stolen Chariot.
This is a short story, set sometime in between The Sea of Monsters and The Last Olympian. It focuses on Percy and Clarisse La Rue's attempt to recover Ares's stolen chariot.
- The Demigod Files: Percy Jackson and the Bronze Dragon.
This is a short story set sometime in between The Sea of Monsters and The Last Olympian. It focuses on Percy and Annabeth's attempt to rescue Charles Beckendorf from giant ants.
- The Demigod Files: Percy Jackson and the Sword of Hades.
This is a short story set in between The Battle of the Labyrinth and The Last Olympian. It focuses on Percy & Co's attempt to recover Hades's new sword.
- The Kane Chronicles: The Red Pyramid.
This is the first book in the The Kane Chronicles series. As near as I can figure, it takes place around now, in between Percy Jackson and the Olympians: The Battle of the Labyrinth and The Last Olympian. It focuses on Sadie and Carter Kane's attempt to stop Set from blowing up North America.
- Percy Jackson and the Olympians: The Last Olympian.
This is set a year after The Battle of the Labyrinth, and is the final smackdown in the Second Titan War.
- The Demigod Diaries: Percy Jackson and the Staff of Hermes.
This is a short story set a month after The Last Olympian, and focuses on Percy and Annabeth's attempt to recover Hermes'caduceus.
- The Demigod Diaries: Son of Magic.
This is a short story written by Haley Riordan, Rick Riordan's son, and is canon. It focuses on what happened to an enemy demigod after the Second Titan War.
- The Kane Chronicles: The Throne of Fire.
This is the second book in the The Kane Chronicles. It focuses on Carter and Sadie's attempt to wake the god Ra.
- The Heroes of Olympus: The Lost Hero.
This is the first book in the The Heroes of Olympus series. It focuses on Leo Valdez, Piper McLean, and Jason Grace's attempt to free the goddess Hera. It is set around six months after The Last Olympian.
- The Demigod Diaries: Leo Valdez and the Quest for Buford.
This is a short story that focuses on Leo's attempt to recover a part before his project destroys Camp Half-Blood, while simultaneously dealing with the Maenads, the crazed followers of Dionysus. It's set about a month after The Lost Hero.
- The Heroes of Olympus: The Son of Neptune.
This is the second book. It focuses on Percy's experience with the Roman Camp Jupiter. It's set around 5/6 months after The Lost Hero.
- The Heroes of Olympus: The Mark of Athena.
The third book. This starts around -5 minutes after the end of The Son of Neptune. It focuses on the Seven of the Prophecy trying to avoid Camp Jupiter, and Annabeth's attempt to recover the Athena Parthenos.
- The Heroes of Olympus: The House of Hades.
This is set two days after The Mark of Athena, and focuses on (spoilers for the end of The Mark of Athena)
Annabeth and Percy's journey through Tartarus,
and Hazel's learning how to use the Mist.
- The Heroes of Olympus: The Blood of Olympus.
Set around a month later, this is the final smackdown with Gaea and the Giants.
The crossovers are set after The Heroes of Olympus, according to Uncle Rick.
- Percy Jackson and the Olympians/The Kane Chronicles: The Son of Sobek.
First crossover. Carter and Percy fight a giant crocodile.
- Percy Jackson and the Olympians/The Kane Chronicles: The Staff of Serapis.
Second crossover. Annabeth and Sadie fight the god Serapis. Set a few months after The Son of Sobek.
- Percy Jackson and the Olympians/The Kane Chronicles: The Crown of Ptolemy.
Third and final crossover. Set a few months after The Staff of Serapis. Carter, Sadie, Annabeth, and Percy
fight Setne, who escaped in The Serpent's Shadow, who's trying to make himself a god.
- Magnus Chase and the Gods of Asgard: The Sword of Summer.
The first book in Magnus Chase and the Gods of Asgard. Magnus and his friends rebind the wolf Fenris. Set a few months after The Heroes of Olympus.
- The Trials of Apollo: The Hidden Oracle.
This is the first book in The Trials of Apollo. Apollo falls to the Earth as a human for punishment, and must fix the broken Oracles.
- Magnus Chase and the Gods of Asgard: The Hammer of Thor.
The second book. Magnus and friends must recover Thor's hammer, Mjolnir, which has been lost.
...And the rest of The Trials of Apollo and Magnus Chase and the Gods of Asgard haven't come out yet, but there are two that I can put in the list.
- Magnus Chase and the Gods of Asgard: The Ship of Nails.
Final smackdown with Loki.
- The Trials of Apollo: The Dark Prophecy.
Second book. Apollo must fix another Oracle.
All information in this answer came from my own reading of all of these books.
The top order has been made depending on how the information has been given, such as the introduction of characters, plot points, etc. This ends up mostly being in the order that Rick Riordan released them in, with a few minor changes that take into account later books.
There are a few stories that I didn't include here, these being Percy Jackson's Greek Gods, Percy Jackson's Greek Heroes, and the stories included in the app.
The Gods/Heroes can be read at any time, as they are the ancient Greek legends, not part of Percy's story. The stories that come with the app aren't necessary for the others, but if you want, read those any time after Percy Jackson and the Olympians, as that's what you need to understand those.
