Chapter 1: Prologue: Rally to Maguuma
Nothing to complicated here: simply don't bother doing anything but run across the desert to your destination.
Chapter 2: Torn from the Sky
Start by running up to the edge of the cliff as shown by marker, then glide down to camp and wait for NPCs to catch up and finish talking. Choose to take Laranthir (the sylvari) to recover the captured soldiers (this route has less dialog). For this next part, ignore the marker and glide across the chasm as a shortcut (see image below).
From there head up to where the sylvari soldiers are being held captive and free them by killing the mordrem. You will then team up with them to take down a champion and head back to camp (feel free to skip all enemies on the way back as you will lose aggro on them). Defend the camp and a bit more RP and you're done.
Chapter 3: Establishing a Foothold
Unfortunately, not much you can skip here. I usually start the conversation with the first NPC and then immediately head to the vendor to sort out my inventory. Once the NPC is done talking, simply use the waypoint (make sure the NPC is done before you use it or you'll have to talk to her again) at the main camp and jump down to talk to the second NPC. While you're waiting for him to stop talking you can head back up to where the waypoint is (don't use it though or you'll have to talk to him again) and glide over to the NPC up on a cliff (there might be another way up, but I don't know what it is) and this will be the last NPC you need to talk to. While this last NPC is talking, you can immediately head over to the entrance of Chapter 4's instance.
Chapter 4: The Jungle Provides
Your party will quickly notice some mordrem attacking some hylek, feel free to head over and attack the mordrem while they give chase as it will cull their numbers and they won't attack you back until the hylek stop fleeing. Once you're done fighting, just deal with the RP and follow the hylek back to their village (tip: try to stay right behind the hylek as they will sometimes stop if you get ahead of them). More RP and then pick to attack the mordrem guard (less RP this route). Your first fight will be in an open field, simply dispatch them all and follow the hylek to the first set of vines. As soon as the vines are cleared, try to get all the enemies out of the circle in the middle and take it for yourself (this way you can save time by claiming the area while fighting the mordrem instead of doing one after the other.
Repeat for the second area. Rest is simple, just kill all the mordrem and postules with a bit of RP and you're done.
Chapter 5: In Their Footsteps
Run over to the marker and talk to the NPC. As soon as dialog commences, start running over to the first scout's location and simply wait for the warbler to appear in your inventory.
Use the warbler, talk to the scout and head over to the second scout.
My route for scouts 1 and 2.
Use the warbler again and talk to the second scout. You can now get rid of the warbler and start heading towards the third scout. I usually use the bouncing mushroom seen in the picture below and immediately glide to the platform seen in the picture below that one. From there, simply run up the ramp and head over to the third scout.
Once this third and final scout is done talking, simply run across the village and bounce out to finish this chapter.
Chapter 6: Prisoners of the Dragon
Initiate RP and walk over to the bridge (but don't cross it completely or NPCs can bug out and not continue the story) and wait for them to finish talking. Once they're done, follow them up the stairs, kill the mordrem you find and proceed to the next batch. Once you've slain all the mordrem in the bivouac, head over towards the tendrils to queue some dialog, turn 180 and head up onto the platform where you'll always find the grenades.
Use the grenades to get past the tendrils and then spam them on the boss and flowers until dead (you can alternate between the two boxes for unlimited grenades). This next part is a bit tough to do without dying (dying isn't a big deal though as you don't have to run back from very far), but you can skip all the mordrem and focus exclusively on the cages to save a bit of time. Then head to the wall so NPCs can break it, suffer more RP and then free Eir so she can die. Kill the mordrem, more RP and leave instance. Talk to the NPC and if your gear suffered death damage, feel free to use the repair anvil right next to the waypoint while you wait for the NPC to shut up.
Chapter 7: Prized Possessions
Run to each point and save the NPCs along the way.
Follow the path shown by screenshots. This will allow you to run ahead while the NPCs RP.
Jump down and when the tool tip updates to "Follow Ruka," run up to the wall for a skippable cutscene. Then, turn around and grab the egg.
Dash (#2 skill) through the thorn wall and across the field (should be able to do it twice before you reach the wall).
Jump up to the second leaf platform (using the #1 skill) and look up to launch yourself with the #3 skill.
Dash through the room and make a left, then use the wall on your left to launch yourself upwards and glide across the chasm.
Dash through another thorn wall and launch yourself in between pods (feel free to use gliding to correct positioning).
Once at the top, dash through the wall that likely hasn't faded yet and glide across another chasm (use dash while gliding to speed things up).
Just dash towards the end and launch yourself to the last platform for more RP to conclude this chapter.
Chapter 8: City of Hope
Ignore the NPCs and run over to read the pedestal (this will skip all their dialog).
Now, glide down to the next location, no need to wait for NPCs, they'll catch up when and where they're needed.
Once at the portal, simply walk back and forth on it until it teleports you to the first trial.
At the first trial, start it up by reading pedestal and then kill all the enemies to move on to the second trial. For the second trial, enter the room and head right to hit the first switch, then go up the stairs and jump onto the walls to skip the labyrinth to a second switch, run back and repeat for the third. This should give you lots of time to clear it even without killing any enemies for more time.
EDIT: This first method has been fixed.
Once you're done reviving the third, head back into the corner where you found the second NPC and pull the regular dog using your 5 skill. By the time the veteran patrols over to you, the regular should be at about 50% HP. Finish it off and enjoy your 4 vs 1 against the veteran.
Turn in the egg, skip the cutscene and use the teleport pad for a bit of RP and you're done.