The best way to do this is to zip your image files. Then, you can unzip them on your new site. Here's how to do that:
1. Select all of the files or folders in the File Manager that you would like to upload to the other site. You can select more than one by holding down your Shift or CTRL key (depending on if you want to select all or only some).
2. Right-click on the selection and choose Download as zip.
3. Download the zip file to your computer.
4. Open your other site and click Files.
5. Upload the zip file and right-click to choose Unzip.
6. All of your files and folders will be added to your File Manager very neatly!
To view a tutorial, visit here: Using the File Manager.