Can I edit shapes in Microsoft Visio Professional 2013 and save them as new shapes? I tried to ungroup the shape so I could edit the shape but Visio does not seem to work the same way as other graphic applications.
I've edited your question as you have confused shapes and stencils. A stencil is a file containing a selection of shapes. A shape is what you drag and drop from a stencil onto the drawing page.
The easiest way is to open the stencil you want to copy from, then More Shapes -> New Stencil. With the original stencil select and copy the shape you want to work with (click the shape and Copy is on the right-click menu). Now go to your new stencil and paste it (again, on the right-click menu). Now you can edit the shape. Don't ungroup shapes. That will break the shape - permanently.