Standard Transwarp Feature

Fast travel in Star Trek Online is achieved by transwarping using the Transwarp feature from the Captain Commands menu on console or clicking on the Transwarp icon on PC.  This feature can only be used once every 15 minutes, but you can reduce the cooldown time by using duty officers and traits, and is available from level 1.

The number of destinations available to transwarp to depends on your fleets level, diplomacy/marauding duty officer level and also your reputation level.  Please check out the wiki for more details.

Fast Travelling

Apart from the standard way of transwarping, you can also transwarp direct to an active mission.  So, if you have a mission all the way across the other side of the beta quadrant, for free (you used to have to pay for it using Energy Credits but this changed in a previous update), you can transwarp straight to the mission instead of having to fly all the way there.  This can save a lot of time, especially if levelling up a new character.

To fast travel to a mission, simply open up the Mission Journal, go to the Active Missions tab, select the mission you want to do and select Make Primary.
Once you have set your primary mission, open the Captains Commands menu, scroll down and select primary mission and select the Transwarp option.  If this option isn't available, select the Travel option instead.
Note:  Missions that are in a different quadrant to where you currently are will have the Transwarp option, but if you are in the same quadrant as the mission you will get the Travel option.

Fast Travel Locations

If you can't transwarp straight to a particular point of interest for whatever reason, you can use the transwarp to mission feature as described above.

Here's a list of useful places around the galaxy and their nearest mission.  Just activate the mission from the Mission Journal, make it your primary mission and transwarp straight there.

Point Of Interest Sector Nearest Mission Story Arc
Andoria / Vulcan Vulcan Sector Ragnarok Future Proof
Archer System Narendra Sector The Core Of The Matter Yesterdays War
Argala System Syllerran Sector Mindscape Delta Quadrant
B'Tran Cluster Vorn / Iconia Sector The Core Of The Matter or House Pegh Yesterdays War / Iconian War
Badlands Cardassia Sector Spoils Of War Cardassian Struggle
Bajor Bajor Sector Jabberwocky Cardassian Struggle
Deep Space 9 Bajor Sector Jabberwocky Cardassian Struggle
Defera Defera Sector Cold War Incoming Hails
Dranuur Colony Antos Sector Melting Pot New Frontiers
Drozana Donatu Sector Time In A Bottle Iconian War
Hodos (shuttle race) Yontasa Sector Capture the Flag Delta Quadrant
Ker'rat / Nukara Boreth Sector Time In A Bottle Iconian War
Kobali Prime Kotaba Sector Kobali Crisis Act 1, 2 or 3 Delta Quadrant
New Romulus Azure Sector Sphere of Influence Solanae Dyson Sphere
Nimbus Hobos Sector Uneasy Allies Iconian War
Qo'noS (Klingon Only) Qo'noS Sector Bringing Down the House Warzone
Sol Vulcan / Teneebia Sector Blood of Ancients Iconian War
Solanae Dyson Sphere Azure Sector Surface Tension Solanae Dyson Sphere
Tzenkethi Battlezone Gon'Cra Sector Mirrors and Smoke New Frontiers

This list will be updated as and when new locations are found