I purchased clipless pedal shoes today and have to remove the cleats from my mom's shoes to mine. However, after unscrewing the cleats I noticed that there are screw-like parts that need to be installed on my shoes. How do I remove these without ruining the threads? These screw-like parts are what I unscrewed the cleats from.

If you are talking about the threaded sections the bolt goes in, then some shoes have them built in, some have a separate plate which you install under the footbed. Have you removed the rubber insert in the bottom of the new shoes? They often have them semi molded on. Check the notes that came with your new shoes.

Nevermind, I had to get new bolts because the older shoes' bolts were built into the shoe itself.