Objective: Break Matt Miller Out
Head to the Broken Shillelagh to start this quest. At the start of the quest, you'll have a top-down view of your tank. Kill the first tank, then the wave of bikers then finally the last set of tanks. This will correct the view afterward.
To proceed, destroy the barrier then go down the ramp. You can use your cannon by pressing RT and your machine gun using LT. Head to the first gate and destroy the targets along the way. Make your way to the next gate where you'll get trapped after Zinyak turns the gate invulnerable. Survive enemy attack until you get off the tank.
Head to the bike marked in your map and ride to the tower before the timer runs out. The path is straightforward and the obstacles are minimal.
After reaching the tower, you'll have to choose whether you'll enter the blue or red door. Entering the red door will end the game quickly but you'll earn an achievement (You Choose... Poorly). To continue with the game, just load your last checkpoint and select the blue door to continue.
Entering the blue door will make you play an old school RPG-style game. The choices are very much limited and you'll end up rescuing Matt nonetheless. After the scene, the mission will be complete.
Objective: Retrieve Matt Miller
Talk to Kinzie to start this mission. Once you're in your power armor, destroy the crates to repair the armor. Next, continue forth until you encounter a door that you can't open. Press A to activate your jetpacks and boost your power armor up. You need to land and let the energy to recharge before going up again. Once you've reached the door big enough for you, press LB to dash and perform a shoulder bash.
Continue forth and take out the guards until you reach Matt. Break the door open to find him in the marked room. Escort him to the next room where he'll find parts to enable the ranged weapons of your power armor. Defend him until he gets all the parts he needs then clear the area for him to install your new weapons.
After the impromptu upgrade, your power armor should be able to fire rockets and cannon. Clear the area and head to the dock. More enemies, including tenacious murderbots will be waiting there. Aim at the ground to take advantage of your cannon's splash damage then continue taking out more enemies. After a while, your guns will stop working so you'll have to wipe the remaining enemy forces with your melee attacks. Once the area is clear, head to the marker to complete the current sequence.
In the next sequence, you'll have to man the turrets and shoot down your pursuers. After a short while, a shielded ship will follow you. Don't bother shooting this as you won't scratch it. Just concentrate taking out the small fries. After reaching the exit, you'll have to wait as Kinzie opens the door. The shielded ship will appear again and will charge its beam. Kinzie will tell you to shoot the left panel but the correct one you should shoot is to the right. This will cause the hatch to close, destroying the shielded ship. The mission will be complete afterward.
Matt Miller will become available as your homie and Loyalty missions will be unlocked as well.