Now, in your web hosting control panel DNS settings for your domain, click ADD RECORD, select CNAME, and in the DOMAIN field add 4321abcd, the first string of ADD_THIS.my.shinynewblog.com, DO NOT add .my.shinynewblog.co.za. Then, in the CANONICAL NAME field (might be called something slightly different dependent on your control panel) add 9876abcd.dv.googlehosted.com (remember to change it to whatever the value of the first string is for your own website), the entire thing including .dv.googlehosted.com. and click OK or SAVE. Adding this value is similar to a house address, so when someone visits my.shinynewblog.com it directs the browser to the address of the files hosted by Google instead of at your hosting company.Secondly, you need to add one more value, which is another CNAME, so ADD RECORD, select CNAME and then in the domain name field add BLOG, or NEWS or whatever you want the subdomain to be, in the CANONICAL NAME field add ghs.google.com, then click OK or SAVE. Now you should have two new DNS records in the list, similar to what is below, just with different values.
4321abcd.my.shinynewblog.com. CNAME 876abcd.dv.googlehosted.com.
blog.myshinynew.com. CNAME ghs.google.com
You will probably need to press one more button to UPDATE or CONFIRM or SAVE or something similar to save changes and then you will have added the CNAME values for the subdomain.
Now, head over to Blogger.com, go to SETTINGS, scroll down to publishing and then click CUSTOM DOMAIN. Enter your subdomain name, save and then if your site uses HTTPS, not just HTTP, then make sure this is turned on.