Tomb Raider

You'll come across optional tombs in the game. They are very short, straightforward, and easily done. You will get these on your map later with a skill you can purchase, but you may as well do them as you go along.

Tomb of the Unworthy

You will see a cage hanging with bodies wrapped up and attached. Jump on this piece and quickly jump to the next part ahead. Light your torch and jump back. Light the bodies on fire. Now go to the other side where there is another cage and knock it off. This completes the puzzle. Now just jump on the hanging cage and wall climb up to the area that has the chest.

Hall of Ascension 

There is a window blowing wind and a platform in the middle. Lower the window panels via the crank under the window. Run to the platform. Look to the window and you should see 3 panels. Once the first one pops open, start turning the crank on the platform to raise it. Quickly get on top of it. When the final panel of the window opens, it will blow the platform close to the wall. Jump onto the wall there and shimmy around to the top part where there is a tunnel leading to the chest.

Chamber of Judgement

When you get inside, you'll see fire and gas cans. There is a statue in the middle back. Get on top of the platform and look at the statue. Turn right and see a tire from an airplane hanging. Use your rope arrow on it to break the platform with a canister on it. Throw all the yellow canisters around up onto the left platform that has a canister already on it (this overloads the platform and it breaks). Now throw three of them on the platform on the right to break it as well. Take all four and put them on the end of the teetering wreckage piece. You can now use this as a ramp to jump to the wall with your axe. That completes the tomb.

The Flooded Vault

You'll get to a part with water and electricity which is making the water dangerous to walk through. You must shoot the barrier with a shotgun across from the raft. Burn the part next to the raft to untie it. Now go to where you use the shotgun and rope pull the raft to you. Now use it to jump to the other side. Pull the top part and let the electric thing go up in the air. Now run across the water. Pull the raft from this new safe point near the electric part. Raise it with rope pull again, and position the raft below it. When it falls this time, the water will now be safe to cross and finish the tomb.

Well of Tears

When you get inside the tomb, you'll see a very obvious puzzle. There is a lift that is higher than you can get to. Throw the canisters scattered around the area on top of it to weigh it down. Get on it and throw the canisters off. Now run up the slope and around to jump on it as it raises, and jump to the next location up. The chest is right up there.

Stormguard Sanctum

There are gases blocking your way after you shoot your fire arrows to get rid of the obstacle. Shoot another fire arrow inside the gas to cause an explosion. Now quickly climb up and continue the path. At the top is a beam. Look across and shoot a fire arrow to block the second obstacle. Now just jump over the gap and you'll find the chest.

Temple of the Handmaidens

You will see a cage. Raise it with the crank nearby and you will see a buoy float under. Quickly jump up to the section on the left. The buoy will rotate the piece there and you can then use the pole to jump over the gap. The next pole has to be moved twice, so you will need to do this part somewhat fast. Rope pull it once and the buoy will take care of the second rotation. Now use it to jump to the rocky section you can use your axe on and climb up.