How to create a widget based on core wordpress gallery widget?

I would like to create a widget based on the wordpress Gallery widget. Where would I find the code to copy and change for this?

I want to change the organization and styling of the output so that captions appear as overlaps to the images and the images have a rounded corner (which I believe means I need to call the images into the background of a div with border radius settings).

All advice welcome, I’m new to creating my own widgets/plugins.