Raising a Stink (Alliance)

For this part of the questchain, queueing for 'The Crown Chemical Co.' won't help. You can't walk back. You need to get to the outside of the instance in Silverpine Forest..

For Alliance, the fastest way to get there is using the toy [item=140192]. It teleports you to Dalaran (Broken Isles). When you go to the center of Dalaran at 49, 48 and stand on the middle stone, it teleports you to Aegwynn's Gallery. When you run to the end (down), there is a portal to Dalaran Crater. Dalaran Crater is in Hillsbrad Foothills what is just next to Silverpine Forest.

EDIT: With a recent hotfix [npc=38066] should now offer a teleport to the Keep while you're on the quest.

To get to Shadowfang Keep from Stormwind as an Alliance:

Mages that have learned the spell can use Ancient Teleport: Dalaran to get right next door (just remember to slow fall!)
Rogues who have obtained it can use [item=139590] and fly the short distance over.
Druids can [spell=293887] to the Hinterlands and fly.
Worgen can use [item=211788] to get to nearby Gilneas.

Everyone else:

If you can find a friendly mage to cast Ancient Portal: Dalaran that'll get you there, just remember to break your fall somehow.

If you've done the pet dungeons you use your [item=140192] to talk to [npc=121602] and teleport to Stratholme entrance and fly which should be a bit faster than flying from Stormwind.

I dont think teleporting to Twighlight Highlands via the Stormwind portal saves much time, so if none of the above applies you'll need to fly it.

For Alliance, I did Boralus -> Ironforge -> Refuge Point - Arathi -> Fly to Shadowfang Keep. Portal in Legion Dalaran no longer available to me.