What is the quickest way to get to the Fleet?

Several ways:

• Quick travel to spaceport (cooldown 30 minutes)
• Use the emergency fleet transport (cooldown 9/12 hours for subs/F2P, reducible to none)
• Use the flashpoint transports (no cooldown)
• Travel to your stronghold, then to fleet(no cooldown)

There is a special ability called “Emergency Fleet Pass” that you can use to instantly teleport to the fleet. It has an 9 hour long cooldown (12 for non-subscribers), although ther are 3 Legacy perks to reduce the cooldown by 3 hours each (so you can potentially have it without a cooldown).

If you have registered a Physical Security Key or Mobile Security Key app (iOS or Android), you can buy a Fleet Pass from the Security Key Vendor outside your capitol city or on your faction's fleet for 1,000 credits.

It has its own 1 hour cooldown, which isn't shared with the Emergency Fleet Pass.

Some planets have a shuttle in their spaceport which can take you straight to it. You will still need to take the elevator for the fleet's space hub to the main area, but this is for characters of any level which have gained access to the fleet from their class's starting area. This is also faster than going to your own ship because you skip a loading screen and messing with the galaxy map.