How to prevent the paper from getting damaged due to a steel compass?
You will need to rest the point of the compass onto some other object first i.e. one that won't damage the paper. 

Some sort of disc

A felt or rubber disc comes to mind. Something thin with enough friction so that the point and disc won't move about. 

You could also just take a cut of eraser and jab it on the point as well. This has a small disadvantage of erasing while you rotate but could have it place in your techniques. 

There is a product called a center disc that also serves this purpose. It has the added bonus of being transparent allowing you to center the point on its target and not damage the paper. 



Image from art-design-glossary.musabi.ac.jp

Clear plastic has the added advantage of letting you see your work and centering your circle easier. So using a sufficiently thick plastic and cutting some small shapes would work just as well. 

If you do make your own practice with them first to be sure the compass does not penetrate it as well. Or enough that it makes the plastic stick out as a point either. 

Consider a different tool

A tangential answer to your question would be to use a plastic flat compass that bypasses this issue completely. 



Image from amazon.ca

This protractor compass has many holes littered along its length. You just need to grab your drawing implement of choice and insert it into the desired hole. Then rotate around like a normal compass.  Pressure from your opposite hand (as seen in the photo from the users left hand) will ensure the tool remains rotating from a fixed point. 

Look at the below zoomed in image of the above tool. You will notice the measurement markings which denotes the precise nature of the tool.



Click image to enlarge