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The second major story mission that Star Trek Online captains are tasked with completing is "Diplomatic Orders." Admiral Quinn should give you this mission over subspace after you complete "Stranded in Space," or you can speak to him in his office to pick it up.
"Diplomatic Orders" has several phases, and requires you to battle foes both in space and on the ground, before eventually leading to your first encounter with the shapeshifting Undine.
The overall mission is not terribly lengthy, and the rewards are more than worth the effort.
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Your Diplomatic Orders
Your first task is to speak to Ambassador Sokketh at Earth Spacedock and escort him to P'Jem to ensure the safe delivery of some artifacts he is carrying. You will find Sokketh in the State Room, standing to the right of the aisle at the bottom of the staircase.
Speak to the Ambassador, then return to your ship and depart to sector space. You'll find P'Jem in the Vulcan Sector of the Sirius Sector Block.
Pretty much just around the corner.
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Space Battle Over P'Jem
Upon warping into the P'Jem Sector, a Klingon captain hails you immediately. He'll accuse you of harboring a shapeshifter onboard your ship, then give you an opportunity to ask some questions before you break off communications.
You should see two Birds of Prey in the sector. Don't get too confident, however, because when you approach one, two more will decloak and attack your ship. Fight them off (even three enemies is no challenge at this level) and pick up any loot that drops.
Don't forget to use your scanners to locate any anomalous data that might be in the area. You can use these items to craft more effective weapons and consumables at Star Trek Online's Memory Alpha.
After both groups are defeated, you'll be prompted to beam down to the planet.
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Battle Klingons on Planet P'Jem
Several groups of Klingons have beamed down to the surface, and it's your job to rescue the planet's peaceful monks before anyone gets hurt. Take any bridge officers you have thus far and a few redshirts down to the planet's surface.
The surface of P'Jem is covered in lush vegetation and distinctively Vulcan buildings. It's also swarming with Klingons.
Take on the Klingons one group at a time, making sure to pick up and equip any useful loot that drops and to empty any flashing storage containers you come across. You need to fight your way up the hill to the abbey.
Eventually, you'll receive a message from your ship that the body of Ambassador Sokketh has been found on Earth. The passenger that has just beamed down to the surface was an impostor. The "Ambassador" will appear before you, then revert to his true form. You won't be able to kill him, but once you've done enough damage he will retreat. Chase the newly-revealed Undine down the hill, fighting off any Klingons you encounter.
At the bottom of the hill, you'll be prompted to scan for the Undine. Doing so will reveal that he's gone, and you will need to return to your ship to pursue another vessel that was hidden in orbit.
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Defeat the Tethys Dreadnaught
Back in orbit, your ship will come under attack by an Tethys Dreadnaught. Survival is the name of the game here, you have little chance of defeating the massive ship without help. Put full power to your shields, shoot down any plasma torpedoes that come your way, and keep your strongest shield toward the enemy. After two minutes, reinforcements will arrive.
Five Federation ships warp in, and with their combined firepower, the Dreadnaught is defeated quickly. After you've taken it down, scan the debris for clues.
Captain Thelin of the U.S.S. Kirk hails you, and delivers a bit of foreshadowing about a future mission. Leave the system when you're ready, then report to Admiral Quinn to complete the mission. Your reward is a nice new shield for your ship, as well as access to two new assignments.
Don't forget this encounter with the Undine, as you will find yourself dealing with them in the near future. Also worth remembering is captain Thelin and the U.S.S. Kirk, as you'll be fighting alongside him again in Star Trek Online very soon.