With the proper equipment, you can hang pictures without damaging your stone wall. You shouldn't attempt to use the same hardware you'd use for drywall. However, your local hardware store should have everything you need. With the right tools for the job, hanging a picture on a stone wall is similar to hanging a picture anywhere else.
Determine where on the stone wall you want to hang your picture. Mark where the top of the picture should be with a pencil or a sticky note.
Measure the picture to determine the distance between the top of the picture and the wire or hook on the back of the picture. Measure this same distance below the existing mark on the wall and mark a spot where the center of the picture should hang. This is where you will drill.
Drill a hole in the spot you have marked. Drill at least as deep into the stone wall as the length of the anchor.
Push the anchor into the hole you have just drilled until it is flush with the wall. This may require a rubber mallet.
Using the drill or a screwdriver, attach the hard wall hook into the anchor with a screw. Hang your picture on the hook, using a level to make sure it's straight.