The Garrison Campaign

After reading the comments I realized that I was also stuck in some of the quest chains, but I didn't know which. Therefore I made the macro below. It will check for the first quest in each line.
If the answer is "true" but you miss the corresponding achievement, you have open quests in this chain.

Two of the garrison campaigns are missing (Primal Fury and Strike at Heart) as they are currently not linked in wowhead.

/run local out="Warlord's Council: "..tostring(IsQuestFlaggedCompleted(35843));print(out)
/run local out="Darktide Roost: "..tostring(IsQuestFlaggedCompleted(35876));print(out)
/run local out="Ring of Blood: "..tostring(IsQuestFlaggedCompleted(36219));print(out)
/run local out="Bloodmaul Compound: "..tostring(IsQuestFlaggedCompleted(34309));print(out)
/run local out="Ours is the Fury: "..tostring(IsQuestFlaggedCompleted(32980));print(out)
/run local out="Sargerei: "..tostring(IsQuestFlaggedCompleted(35185));print(out)
/run local out="Iron Siegeworks: "..tostring(IsQuestFlaggedCompleted(34034));print(out)
/run local out="Heart of the Breaker: "..tostring(IsQuestFlaggedCompleted(34335));print(out)
/run local out="Fall of Shattrah: "..tostring(IsQuestFlaggedCompleted(35679));print(out)
/run local out="Broken Precipice: "..tostring(IsQuestFlaggedCompleted(36117));print(out)

If you are wondering how the weekly quest spawning works exactly, I have done a lot of research into it. I have a lot of alts and for some reason I decided I wanted to get this done on all of them. Here are my conclusions.

Normally, you only get one quest per week. Sometimes, however, you get two quests the same week. This is guaranteed to happen for two of the quests:

• The same week you complete the last quest in the first set of quests you also get the first random quest in the second set (so you will get the 4th and 5th quests the same week).
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• It is also guaranteed when completing the second set of quests; when you complete your 8th quest you will always get Garrison Campaign: Deep Recon the same week.

You may also randomly get an extra quest the same week, but you will never get more than two quests the same week.

If you are eligible to get a second quest for the week, it will usually appear immediately after a relog. So if you want to get the whole chain done as fast as possible, relog and see if you have another quest waiting for you!

If do you have a quest waiting for you, it will usually show up on your map as a yellow ! over your Garrison. However, a few of them do not, so don't rely on that.

Quests appear weekly in the following order:

First Set
You get one or two of these four quests each week. The order is completely random, but you won't get any other quests until all four are completed:

• Garrison Campaign: The Sargerei (Reward: [item=119134])
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• Garrison Campaign: Darktide Roost (Reward: [item=119130] via mail)
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• Garrison Campaign: Grinding Gears (Reward: [object=234425] outside your Garrison)
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• Garrison Campaign: The Ring of Blood (Reward: [title=397] title)

Second Set
Works the same way as the first set. You get one or two of these each week and the order is random:
Note that you do not have to wait a week to get the first random quest

• Garrison Campaign: The Fall of Shattrath (No special reward)
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• Garrison Campaign: The Search for Shadow Hunter Bwu'ja (Reward: [item=119151])
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• Garrison Campaign: The Broken Precipice (Reward: [item=119128] via mail)
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• Garrison Campaign: Farseer's Rock (Reward: [item=119145])

Final Set
From here, the final four quests are given in a linear order:

1. Garrison Campaign: Deep Recon (No special reward)
   Note that you do not have to wait a week to get this quest
2. 
   
3. Garrison Campaign: Crows In The Field (Reward: [item=119139])
4. 
   
5. Garrison Campaign: Missing Grunt (No special reward)
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7. a. Garrison Campaign: Awakening (Reward: [follower=217])
   b. Primal Fury (Reward: [item=120311] and [item=120313])

Note that the bonus objectives that come with each set are completely optional - you don't have to complete them to progress. I recommend skipping them unless you need the extra Apexis Crystals!

Seems like we're getting a Garrison campaign quest every week. Unrelated to which dailies we have or haven't done, this is likely a rotation over 12 weeks. They require level 100 to show up. Each chain rewards 800 [currency=823] on completion of the objetive bar you get while in the zone.

- Darktide Roost -

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• Garrison Campaign: Darktide Roost/Garrison Campaign: Darktide Roost
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• Darktide Defender & Shredder Suppressor
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• Dirty Rats
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• Releasing Rylak & Downsizing the Darktide
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• Rylak Rescue (After completion you get [item=119130] worth 250 [currency=824] on your mail)
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- Ring of Blood -

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• Garrison Campaign: The Ring of Blood/Garrison Campaign: The Ring of Blood/Garrison Campaign: The Ring of Blood/Garrison Campaign: The Ring of Blood
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• How Tough Are You?
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• Entry Fee
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• The Champion of Blood (Also rewards [title=397] title)
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• Out of the Shadows/Out of the Shadows
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- Iron Siegeworks -

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• Garrison Campaign: Grinding Gears
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• I Know The Drill!
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• Maker's Mark & Four Finger Discount
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• A Fellow Gearhead
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• WANTED: Overlord Blackhammer
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• Payment Due for Services Rendered
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• At the end, you're rewarded with [object=234425] right outside the main entrance to your Garrison. On use, it sends you to your Gorgrond outpost.
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- The Sargerei -

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• Garrison Campaign: The Sargerei
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• Infiltrating the Sargerei
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• The Word of Socrethar + Shattering the Enemy + Pesky Podlings + An Ancient Threat
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• Facing Demons
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• Socrethar's Fury > Awards [item=119134]
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• Forging Ahead
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- Ours is the Fury -

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• Garrison Campaign: Farseer's Rock
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• Eaters of the Elements
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• In the Shadow of Giants + Eatercology
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• The Ascent
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• Fury of Frostfire (Rewards [item=119145], currently bugged, you auto-learn it on completion and also get the toy, it says it's already learned)
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• Ours is the Fury
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PD: Apparently some people are getting 2 weekly questlines.