As of the 5.10 Update on July 24, 2018, the previous lists of quests/missions required to progress through Canny Valley have been converted into optional side quest missions and with the addition of a partial story line for Canny Valley, has been replaced by all new missions. EDIT 7/1/19:
There have been a number of updates since the one on 6/24/18, and the requirements for the achievement have changed again since Canny Valley now has a complete story line. New/Updated solution added.
REMOVED 7/28/18: Old quest list and achievement requirements.
REMOVED 7/1/19: Previous requirements as of 7/24/18 update.
has confirmed in the comments, you need to complete all of the Canny Valley missions to get the achievement. You can still complete Storm Shield Defenses 1-6 back to back and then the Launch the Rocket mission to progress to Twine Peaks without doing all of the Canny Valley missions, but you will not earn the achievement.
The Canny Valley story line consists of 19 pages for a total of 69 quests, this includes Storm Shields 1-6 and Launch the Rocket. As you complete each quest a green check mark will be placed next to each quest to show completion. They must be done in the listed order. Any Storm Shields/Launch the Rocket quests you have completed prior are retroactive and you will be given credit for them when you reach that particular quest in the list.
The vast majority of quests require you to find/collect X item in a certain level of zone. You are not confined to any particular mission type, so you can play whichever ones you want. Though at least half of the quests require you to complete them in a Ghost Town zone and depending on what's loaded on the map, you may be confined to 1 or 2 mission types.
You can view the current quests, requirements, and rewards related to Story progress at the wiki here:https://fortnite.gamepedia.com/Canny_Valley_Quests
Please inform me of any additional updates or changes that you feel may be needed to solution in the comments.