There's two checklists here, one daily and the other weekly - thought it might be best to split these up into separate ones so people can get a better handle on the info and not get confused.
Korthia Hub Dailies You'll get 3 daily quests to completefrom the Korthia hub each day, that can reward an assortment of Death's Advance and Archivist's Codex rep, iLvl 200+ catchup gear, Legendary Korthite Crystals and some Anima. You'll mostly want to do these for the reputation rewards above all else.
Korthia Rares, Events and Treasures During your daily rounds of Korthia doing the daily quests you should make sure to get a decent amount if not all of the Rares, Events and Treasures done from across the zone.
<aside> 📌 There some very handy WeakAuras you can get to greatly help with these daily things to do, that we will link in the video description!
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Additional completions of Tormentors of Torghast Event
You can get 100 Death's Advance reputation and 75 Soul Ash for the first time you kill each of the different Tormentor's of Torghast, so definitely worth doing each of the different events.
Ve'nari + Maw Dailies + Maw Rares
If you still need Stygia (for Korthia catchup gear or Shard of Domination un-socket item), then you should do all the dailies and clear the map of at least all the easy rares while you're there. You'll also net a ton of Ve'nari rep too, which can either help you towards getting Appreciated (Exalted) standing or work towards Paragon Boxes (which can contain a cool Maw Horse Mount)
<aside> 📌 Be sure to pickup the weekly quests from both your Covenant Sanctum and Brokers in Oribos before you do any form of content that week.
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This is all the stuff you should be doing weekly while out in the open world, and is mostly done in Korthia with a little in The Maw.
Chains of Domination Campaign Be sure to keep up with the new Chapters of the Campaign that will be available each week, some of which will require you to complete all your weekly Renown tasks first.
Shaping Fate Quest This is your top priority to complete during the week, and it's not something you really need to rush through - unless you want the Stygian Embers and Renown Rewards before raid.
You'll be able to fill this bar up fairly effortlessly if you do the other weekly and bi-weekly events and then just a few daily things sprinkled between.