Chapter 1 - Anchored
Once Marcus is up, you have your first chance to find a collectable. Before even leaving the room, look around. You should find a $20 bill to pick up. It's the first of three collectables in Anchored. The second collectable is mere seconds away. In the hallway, turn right and check the room to your left. You should find Dom's psyche report on the table.
Go to the room indicated on your screen and you'll have a little scene with Dom. Once Dom is following you, check the next room down the hall. You should find a Requisition Form on the table. It's the third and final collectable of the chapter.
You're introduced to Points of Interest here, showing you an empty Lambent stalk. Those aren't ominous at all, certainly you can ignore them. Instead, take a look at the game table in the middle of the room. Want an Onyx medal? Interact with it 2,000 times and you'll earn it. That'll take a while, so maybe you can stop around 1,500 and hope to get 500 more as you play the game.
Time to meet Jace. Get him to stop obsessing over a decade-old pack of mints and he'll follow you to CIC. Meet up with Anya, decked out in COG armor, and learn that you're here to meet up with Prescott, who ran out on you a year and a half ago. What does that jerk want? Guess you'll find out.
You didn't think it would all be this easy, did you? Soon, the Lambent will attack and it's your job to blast them while you make your way to the deck. Once you're there you get to blow up some bigger Lambent and make your way to the deck controls. Flip the switch and the Raven can land. Prescott strides in and takes command as fast as he can, and hands a disk to Fenix. On the deck you can find a box marked with the COG logo. There's a body behind it that, if you shoot it, it drops COG tags for you.
On your way to check out the disk, more Lambent show up. Be careful if you're going to blast them with the shotgun, since they explode and can be pretty painful. Take them out and make your way up to view the message. Marcus, it's your father! He's alive! So that's what you'll be doing in Gears 3 -- tracking down Marcus Fenix's father, trying to get his 'solution' to the Lambent, and dealing with the Locust as you go.
Now you have to make your way to the captain's cabin and rescue Prescott. They introduce you to spotting here, a new feature. Be sure to use it a lot -- it points out enemies for your allies to target. It's also something to do to get an Onyx Medal. Fight your way to the captain's quarters and rescue Prescott. The chapter ends when you find him.