Exit through the wooden gate you just opened to emerge on a ledge above the water.
Below and to the left is a waterfall. Drop down onto the flat block below the doorway.
Take a standing jump onto the sloping ledge beneath the waterfall. Walk through the falling water into the cave. This is SECRET #1.
Walk to the edge and kill the lizard lurking below. Drop down and pick up the MP5 clips. Then climb back up to the opening.
With the falling water behind Lara, walk to the edge and take a standing jump forward to grab the handholds on the ceiling.
Monkey swing around to the far right corner of the cave and drop down in the doorway.
Save the game if you can. Then slide down the ramp almost to the bottom and jump over the flames to grab the edge of the nearest pillar. Pull up.
The middle pillar, with the save/power-up crystal, ignites when Lara passes over it. To avoid touching it, take a running jump from the first pillar over the second...
...to grab the edge of the third pillar. You'll get the crystal in the air.
Now pull up and take a running jump to the ledge below the doorway.
Turn around and kill the poison-breathing lizard below or just move on.
NOTE: If Lara does catch fire, either reload or jump into the pool to douse the flames. Then swim to the ledge below the first pillar, climb out of the water, pull up onto the first pillar and continue from there. Do the same if you fall into the water while crossing the room or drop down off the third pillar, since you can't reach the exit from below.