To change the product group for your category on the Update Product Group page, you can edit the Category ID field to add your new category, then Save.
To change the product group for your category using the Bulk Upload tool, replace the existing Category ID value with a new Category ID value. Make sure that the new Category ID value is valid and copied from the "Category ID" tab on the Bulk Template. Make sure that the Category ID value is the same for all products in the product group.
In either case, there may be attribute changes required. Since attributes are dependent on the category, changing the category may alter the attributes that can be associated with the product or that need to be required. A full list of available and required attributes is available on the "Attributes" tab of the Bulk Upload Template. For instructions on changing attributes, see I Need to Add or Change Product Attributes.