I bought a domain from Google, and my blog is now invisible. Obviously, this is another swindle by Google, stealing my money!These blog owners have not done any research, about the custom domain migration process, or about any issues related to renaming the blog. The successful migration process starts with getting the blog, under its new URL, indexed by the search engines.
One of the most useful tools, that you as a blog owner can use, is the sitemap.
The default sitemap, for a new blog, indexes the most recent 26 posts. For most blogs, you will seldom publish more than that between each indexing pass by the search engines - and a 26 posts submission is generally sufficient.
When you re publish your blog under a new URL, you want all posts reindexed immediately. To get all posts reindexed, you are going to need a more robust sitemap complement.
You want your blog, under the new URL, to be visible under that new URL as soon as possible, so setup a proper sitemap - then get to work publishing new posts, and increase search engine activity. And, use Jump Break for all new posts, to focus search engine attention on the individual posts.