You will have to deal with several simple jumps in just a few seconds. The main objective is to jump from one chain to another (#1). You should be able to reach an upper platform in a few seconds. There's a single chain nearby (#2). Use it in order to get down to the lower level safely (you will have to slide down). I would recommend that you use your flashlight frequently.
Head on to the next checkpoint. There's a simple lesson of swimming ahead of you. In order to dive down you will have to press the F key. If you want to go back up you will have to use the SPACE key. Start off by swimming forward (#1). Thankfully, you will be able to catch your breath along the way. The main objective is to reach a small cavern (#2). There's a lot of light here, so you shouldn't have any problems locating this corridor. You should be able to reach the shore in just a few seconds.
Be careful! One of the nearby corridors is being guarded by a jaguar (#1). I would recommend using the machine gun during this battle, mostly because the animal is very strong. Furthermore, you should be jumping a lot. Thanks to that, the jaguar won't hit you with its claws. Once you're done with that problem, start moving forward. Be careful, though. There's a new trap nearby (#2). You will have to wait for the right moment to start running, so you won't get smashed by the huge walls.
There are TWO identical traps nearby (next to each other). Thankfully, they won't pose too much of a threat. You will have to use a metal crate. It's located on your right. Hold the E key in order to push it (#1). Turn left and start moving forward. The walls will be hitting the box instead of Lara (#2). Thanks to that, you should be able to pass through this corridor quite safely.
Keep moving forward. You should be able to reach a much larger room in just a few seconds. This is where you will have to solve the first puzzle of the game (#1). Thankfully, it's not that hard. Nevertheless, I've explained what exactly has to be done. I would recommend that you explore the entire room first. Here's what you can/must do:
1) kill the second jaguar; the animal is guarding the lower level of this room (#2); you shouldn't have any problems with it, especially since the jaguar won't be able to jump to your floor
2) pick up one of the hidden bonuses (it's on your right)
3) discover three metal plates; stepping on them will activate parts of the main mechanism
4) use the binoculars; pressing the E key will allow you to scan all of the surrounding (interactive) objects; they will be divided to four different categories (mechanisms, movable objects...)
This puzzle is a rather obvious one. Sadly, you won't be able to get past it right away. A few important things has to be done first. The main objective will be to place three metal crates on the tiles located on the floor (#1). The main problem is that only one crate is on this level. The remaining two are on the lower platform. You will have to transport them here. Get down to the lower floor. You might want to consider scanning all of the surrounding objects. You should be interested in one of the movable platforms (#2). It will come in handy in just a few seconds. Start off by pulling a single crate that was underneath the platform. Thanks to that, it will become fully interactive.
As you've probably suspected, you will have to catapult both crates, so they will land on the upper level. Sadly, it's not that easy as it sounds. Start off by choosing one of the crates. You will have to place it on the movable platform, HOWEVER you can't push it all the way. Otherwise the crate would immediately slide down. Only the small part of the crate must be touching the platform (#1). You can go back to the upper floor. There's some stairs on your left. Go there now. You will find yourself on a small pedestal (#2). The construction should look exactly as I've shown on the screen.
There's only one thing that has to be done right now. You will have to make a jump to the opposite end of a movable platform. Thanks to that, the crate will be catapulted. If you're lucky, it will land on the upper level (#1). Repeat these actions with the remaining crate. Now you will have to place them on the nearby buttons (#2). It doesn't matter where they go, because they are all identical. Each time a crate is placed you will unlock a part of the main mechanism.
Once all three crates have been placed a short cut-scene should appear on your screen. You will have to reach the exit. Sadly, it's located on the top floor. Turn left. There is a single chain over here (#1). You will have to use it in order to reach a nearby ledge (#2). Start moving towards the right edge of the screen. Once you're there, climb up to the new platform.