There’s plenty of advice out there on how to automatically move from Blogger to your own site.

The general steps you need to take are:

1. Setup your own site with a hosting provider.
2. Backup your blog site.
3. Import blog posts, the WordPress blog has an import function which works well.
4. Setup Blogger to redirect.
   This was the most difficult thing to find out about.  You want to setup your blogger blog to redirect automatically.
5. Done!

These three sites are the best references I found:
How to redirect Blogger Beta to WordPress.com – October 2007
Moving from Blogger to WordPress: Best Practices – May 2006
Migrate from Blogger to WordPress without Losing your Google Search Rankings (September 2009)