Hanging a picture or painting on your stone fireplace is the same process as any project that requires you to drill into brick or stone. Though not essentially different from drilling into wood or plaster, it does take some specialized equipment. Also, because replacing a brick or stone is a much bigger deal than patching a hole in your drywall, the whole project has a narrower margin for error that can make novices uneasy.
Hold your picture at several positions against your stone fireplace. Choose the location you want it to hang long term.
Locate the point on the back of your picture where it will hang from your screw. Scratch the corresponding point on your fireplace with a drill bit to mark the position.
Beginning with your smallest drill bit, drill a hole at the marked position.
Move up to your next smallest bit and widen the hole you've drilled. Continue to widen until you've drilled a hole with a bit 1/8 inch narrower than the diameter of your masonry screw. This process reduces the chances of chipping your masonry.
Drive in your lag screw with your screwdriver bit, leaving 1/8 to 1/4 inch extending out from the wall.
Hang your picture.