Shift your front derailleur to the smallest chain ring and remove the chain from the chain ring using your hands. Rest the chain against the frame.
Loosen and remove the crank bolts on the crank arms using your 8-millimeter Allen wrench. Thread the crank puller into the holes where the crank bolts were, then use the lever on the crank puller to remove the crank arms from the bottom bracket.
Remove the non-drive or left side of the bottom bracket by unthreading the cup using the bottom bracket tool and a crescent wrench. Do the same thing to the drive or right side but unthread to the right. The drive side is reverse threaded.
Clean out the inside of the frame's bottom bracket shell using degreaser and a rag. Put fresh grease on the threads in the frame and the bottom bracket's threads.
Install the bottom bracket in the frame using the bottom bracket tool and the crescent wrench. Starting with the drive side, tighten the bottom bracket to the left until firm. Then install the non-drive side cup by threading to the right until snug but not as tight as the drive side.
Push the crank arms back onto the spindle and thread the crank bolts back onto the spindle using an 8-millimeter Allen wrench. Tighten the crank bolts until firm. Place the chain back onto the small chain ring.