When you reach the first slope, roll over it and jump at the edge to grab the opposite ledge. Press the Roll button before sliding down.

(Thanks Treeble)

After climbing on the ledge, jump before reaching its edge. That way you will directly grab the lower narrow ledge ahead and traverse to the left around the corner.

(Thanks Treeble)

When you turn around the corner, jump to grab the decrepit ledge. Do not shimmy to the right. Instead, climb on the decrepit ledge and jump to grab the stone block ahead.

(Thanks Treeble)

After the upwards slope and the pole that collapses, and as soon as you grab the ledge, press the Down button and jump to use the Grapple and catch the shimmery bell above. (You do not have to climb on the ledge.)

(Thanks Treeble)

^ MAJOR SHORTCUT ^

You can take a major shortcut and skip the whole mountain part. After exiting the cave, approach the edge and look down to spot the plane's nose. Aim the right hand part of it (where the X of the first screenshot shows) and draw your Pistols. Jump down and shoot whilst in midair.

You will land next to the fan. Crawl beneath it.

Use the Grapple and pull the broken part of the plane. Do it from here, so you will not have to deal with the jaguar once you drop down. When you do the latter, enter the plane and pull the object inside.

Found by: Gandarufu - Watch a video here.

If you cannot do the above shortcut and decide to follow the normal way, continue to the right and jump to grab the stone block. Then roll over the slide and clear it faster. Do not forget to press the Roll button just before reaching the edge of the slide.

Once you reach the ledge above the plane's nose (where the helicopter passes above Lara), jump to grab the plane's wing and release to drop down. You will lose some heath, but this is quicker than climbing down the snowy ledges behind.

(Thanks Tomb Raider Master)

Icy cave: When you reach the part with the two stalactites, grab the first one and jump directly to the platforms that falls down.

Icy cave: When you reach the ledge where you have to jump and grab the pole behind, jump from the top ledge and not the one below.

(Thanks Lara Croft Fan)

When you meet Rutland's men, do not bother with them. Run straight ahead and climb on the snow bank. Jump to the next one ahead, then to the one on the right where the guard with the Grenade Launcher is. Use the Grapple to make it across, grab the wire and slide beyond the pool.

If you cannot avoid Rutland's men, kill them. Then do not climb on the middle snow bank. Jump on the barrel close to the second snow bank and jump up to grab the latter. (Since you will use this barrel, make sure you do not explode it during the fight. Also, when you climb on the barrel, it will not explode.)

(Thanks Pipolinne)

When you reach the first cave with the cold water, do not bother shooting the icy ledge above. Instead, jump towards the next ledge. Lara will land in the water, so quickly climb out of the water and onto the ledge.

Cave with floating icy ledges: Jump into the water and press the Interact button to swim faster. If Lara's health is ready to drop to zero, take a Health pack. Climb on the first icy ledge, then jump to the next ones until you reach the Monastery's entrance.

When you enter the Monastery, ignore the jaguar and jump to grab the first shimmery bell. Make your way to the other side.

Scale puzzle: Kill the jaguar. Jump on the left scale and pull the gold box.

Climb on the small building and use the Grapple to bring the gold box towards you. Do several small steps backward while Lara keeps pulling and you will end up pulling the box to the top platform straight way. You can watch a video here.

Found by: Gandarufu

Alternative way by Tomb Raider Master: Kill the jaguar. Jump on the left scale and pull the gold box. Climb on it, then climb on the building.

Go to the other (right) scale. Lara's weight is not enough to bring it down, so you need to keep jumping on the scale in order to do so. When it is on the floor, immediately grapple the bigger of those two boxes and pull it as far as you can on the scale.

Then detach, climb on it and grapple/pull the smaller one. Then jump to grab the building and pull up. Grapple the gold box which should be lifted up on the other side and pull it.