You should upload the largest size image that you intend to use on your store pages. ShopSite can resize the images down but not make an image larger.
To configure the image sizes you go to Images > Configure and set the various sizes that you want to display. How big to set them is a matter of personal choice but typically you would want the Medium size image no bigger than 350x350, the Small image 150x150 or so and the thumbnail less than 100.
Once you have the resized image sizes configured you should select all your images and click the Create Resized Images button. (note you must select the images in each directory separately, there is no way to do all directories in a single action)
Once that is done the publish should be able to find the appropriate image size. If you are still getting a "Cannot open file for writing" error the permissions are not correct for your images or directories. The directories the images are in need to be writeable for the webserver user the store is running as. If you used FTP to upload the images then you will probably need to change the owner and/or permissions on the files so the webserver user has proper access permissions.