Weathered Pamphlet

If you're wondering about how to get to Gadgetzan really fast as Alliance...

I ported to Stormwind and flew to Cata portal area and chose Uldum, flew from there to Gadgetzan.

TL;DR:
Stormwind -> Uldum -> Vendor Flight to Gadgetzan.

Or, as Kathlynn recommended:

Old Dalaran and New Dalaran have portals to Caverns of Time (should have this hearthstone still)

These old portals are removed as of 8.1.5.

UPDATE!

Due to the redesign of the Portal Room in Stormwind, Caverns of Time has its own portal now. That is the fastest way.

Alliance: Quickest way now is to take the portal to Caverns of Time in the Stormwind mage tower portals area. You come out in Gadgetzan, just a quick hop away from the quest turn in at /way 52.1, 30.1.

if you get any of these quests from islands, do NOT follow the instructions in the quest text. You will be going in circles fruitlessly into dead ends.
Go straight to google/WOWhead.
Sadly, they're all like this.
If I knew how convoluted it would be it wouldn't have been near as bad - I don't mind longwinded adventures with vague objectives as long as they are clearly that with no misdirection (unintentional or not) - but with the new directions on the quests feature it makes for a huge clusterf(#*k.
Horrible, horrible afterthought of quest design basically forcing you to not trust Blizzard and rely instead on Wowhead. Which is very rare.
Even Vanilla wasn't this bad. And that's from experience, and that's saying a lot.
Thank you for "rewarding" us for doing Islands, Blizzard.