The first question people always ask is, where do I start?
This article gives a few good starting points, but with Magic Origins, that set the origin story for the main Gatewatch characters, Jace, Chandra, Liliana, Gideon, and Nissa.
The Comic "The Raven's Eye" is an alternate version of Liliana's Origin. (2023 archive link)
Your best bet is to start reading there, to learn each origin story.
Further Backstory:
The next installments all tell backstory that leads up to the start of the main Gatewatch storyline. You don't need to read these, but they have a lot of specific points of story for some of these characters. They are presented, in timeline order, however, for the main Gatewatch saga, skip to the next section.
The Comic "Flight of the White Cat" covers Ajani's sparking. (2023 update: link)
Next in the timeline is the novel, Alara Unbroken follows Ajani's sparking, and his battle for Alara with Nicol Bolas. He also encounters Sarkhan Vol and Elspeth Tirel.
The Comic "The Seeker's Fall" covers Tezzeret's sparking. (2023 update: link)
The Comic "Honor Bound" shows Elspeths reaction to the shards of Alara recombining, and tells of her origin and leaving of Alara. (2023 update: link)
The Novel: Agents of Artifice - a novel following a younger Jace and Liliana as they work with the Infinite Consortium, then lead by Tezzeret occurs roughly around the same time.
The Comic "Fuel for the Fire" covers Chandra's encounter with Jace, which occurs at some point in the early part of the time during Agents of Artifice. (2023 update: link)
The eStory "The First World is the Hardest" is the origin of Ob Nixilis. (fixed link)
The Comic "The Hunter and the Veil" covers Liliana finding the Chain Veil, and encountering Garruk. (2023 archive link)
The Comic "The Wild Son" continues Garruk's hunt to find Liliana to break his curse, and shows his origin. (2023 archive link)
The Comic "The Veil's Curse" follows Garruk's continued hunt for Liliana, where he encounters Jace. (2023 archive link)
The Novel The Purifying Fire follows Chandra and Gideon. After being defeated by Jace, Chandra encounters Gideon, and they have an adventure together.
The Novel Test of Metal takes place after Zendikar's plot below, but starts before Alara's plot. Much of it isn't considered canon anymore. It was referenced as a fever dream by Tezzeret while being rebuilt (2023).
From here the story diverges. The story next goes to Zendikar, while some characters wait to journey to Mirrodin, while others meet in Ravnica. Garruk and Liliana head to Innistrad.
Zendikar 2009
The Comic "Journey to the Eye" follows Chandra after the novel The Purifying Fire on her journey to Zendikar.
The Comic "Awakenings" follows Jace on his path after Chandra to Zendikar.
The Novel The Teeth of Akoum follows Nissa meeting Sorin Markov and their path to the Eye of Ugin, releasing the Eldrazi, and is considered a terribly written book not recommended to anyone to read.
The Comic "Enter the Eldrazi" follows Jace and Sarkhan as they leave Zendikar, one seeking help, the other, answers.
The eStory "The Battle of Fort Keff" tells the story of Gideon following Chandra's trail to Zendikar, and realizing the Eldrazi are an interplanar threat.
The video game Duels of the Planeswalkers 2014 covers Chandra's chase after her contact about the scroll to Zendikar.
Mirrodin 2009-2010
All of the lore, in order, for Scars of Mirrodin's story can be found in this article.
The Novel Quest for Karn follows Elspeth, Venser, and Koth as they battle through Phyrexia. This novel is largely not canon anymore, and is considered a terribly written book not recommended to anyone to read.
Innistrad 2010-2011
All of the lore, in order, for Innistrad's story can be found in this article.
The eStory "Veil of Deceit" covers Liliana's further attempt to escape the Chain Veil.
The eStory "Beast" covers Garruk's attempts at fighting the veil curse.
The eStory "The Hunter Cannot Pity" covers Vronos, and his journey to stop Garruk, who is hunting indiscriminately.
The video game Duels of the Planeswalkers 2015 covers the story of Garruk's hunt.
The eStory "Monster" concludes Garruks' story from Duels of the Planeswalkers 2015, with Jace leaving Garruk alone, and Liliana escaping.
Ravnica 2012
All of the lore, in order, for the Return to Ravnica story, can be found in this article.
The eNovella The Secretist follows Jace as he uncovers the secrets of the Implicit maze of Ravnica.
Theros 2013
All of the lore, in order, for Theros' story, can be found in this article.
The eNovella Godsend follows Elspeth's journey after leaving Mirrodin on Theros.
From here, all the stories were posted online as articles or video, and so were all covered by guide articles, listed below.
Tarkir 2014
All of the lore, in order, for Tarkir's story, can be found in this article.
A synopsis of these events with links can be found here.
If you skipped the backstory elements, here is where the Gatewatch comes together, and the stories from the past all tie together for Gideon, Liliana, Jace, Chandra, and Nissa, among other planeswalkers.
Battle for Zendikar 2015
All of the lore, in order, for Battle for Zendikar, can be found in this article.
The Gatewatch comes together to fight the Eldrazi.
Shadows over Innistrad 2016
All of the lore, in order, for Shadows over Innistrad, can be found in this article.
Jace goes to Innistrad to follow a lead on mysteries there for the Gatewatch to solve.
Kaladesh 2016-17
All of the lore, in order, for Kaladesh's story, can be found in this article.
The Gatewatch are asked to come to Kaladesh, but what they find there is a revolution.
Amonkhet 2017
All of the lore, in order, for Amonkhet's story, can be found in this article.
The Gatewatch head to Amonkhet in an attempt to stop Nicol Bolas.
Ixalan 2018
All of the lore, in order, for Ixalan's story, can be found in this article.
Jace goes to Ixalan.
Dominaria 2019
All of the lore, in order, for Dominaria (and backstory) and Core 2019, can be found in this article.
The Gatewatch gather together in Dominaria, to regroup.
Guilds of Ravnica 2019-2020
The story continues to this day.