droumerda:

This version is from 2016-05-24
Click here for the most up to date version of this post

Character bios

A great place to start, it will give you a solid base to stand on going
further. Recommended reading order:

Reinhardt
Torbjörn
Mercy
Winston
Tracer
McCree
Pharah
Widowmaker
Soldier: 76
Reaper
Mei
Bastion
Zenyatta
Hanzo
Genji
Roadhog
Junkrat
Zarya
D.Va
Symmetra
Lúcio

In-world news stories

Experimental weapon stolen from Watchpoint: Grand Mesa
Fading glory: On the trail of Jack Morrison (The rise and fall of Overwatch)
Lúcio brings music to life with Syneasthesia World Tour

A Moment in Crime Special Report: “The Junkers”
Mei’s adventures

LumériCo vision for Mexico unvieled, but threats to safety and security remain

Origin

stories

Soldier: 76

Comics

Train Hopper (McCree)

Dragon Slayer (Reinhardt) 
Going Legit (Junkrat and Roadhog)
A Better World (Symmetra)
Mission Statement (Pharah)
Destroyer (Torbjörn)

Animations

Cinematic Trailer 

We are Overwatch  

Recall (Winston and Reaper)
Alive (Widowmaker and Tracer)
Are You With Us?

Launch Celebration Trailer 
Dragons (Hanzo and Genji)
Hero (Soldier: 76)

Congratulations, you should now be up to date with Overwatch lore.

You want more?

Lorewatch (the playlist is backwards, so you have to start from the bottom)
Shortwatch  (In depth analysis of every Overwatch animation)
Comicwatch (In depth analysis of every Overwatch comic)
Overwatch wiki
Google spreadsheet (Timeline etc.)

Bonus:

Heroes of the Storm – Tracer Trailer

Have I missed a source? Please let me know.