First of all, make a backup of all your data in your database and your customized themes and other files.

1. On your new domain, create an index.php, with the function header() , to let the search engine know, that something is coming soon, visit us again.

   <?
   header("HTTP/1.0 503 Service Unavailable");
   header("Retry-After: 3600");
   ?>
   

   Alternatively you can use the Plugin Wartungsmodus or Maintenance Mode.
   From now on you can work, without to be worried about search engines, on your website.

2. After that, you import your data into the new database and upload all files.
3. Adjust the wp-config.php accordingly. Open the file in your editor and add following data:

   define('WP_SITEURL', 'http://www.example.com');
   define('WP_HOME', 'http://www.example.com/blog');
   

   Tip:

   I suggest to this in every other blog too. You save a request to the database and it improves the performance of your installation. Please check the correct definition, even a wrong / can deny access to your blog for visitors.
   After you add these definitions, don't forget to check off both fields in your settings area. See screenshot:
   

4. Now the installation of WordPress starts with the same database files. The new address get pulled out of the both constants.
   Note: if there is a Plugin or modification to cache active, please deactivate it.
5. After the new blog is running, install the Plugin Search & Replace, which helps you to find in old posts the old URL and replaces it with the new one.
6. Now check all preferences in your blog, do it accurately.
7. Everything OK? Then delete the index.php which you created in the beginning. Searchengines are having access now.
8. Add to your old blog an .htaccess file and put in:

   Redirect 301 /blog/ http://www.example.com/
   

9. Now you can delete all files and the database on your old blog.
10. If you have an account on Feedburner, you have to change the URL to your new blog, so your current reader will still get your feeds. But don't change your original feed!

Bonus Tip:

Acquire just your domains at another cheap but reliable provider, where you can change the DNS settings easily whenever you change to a new host. So you won't have any troubles, if your old provider don't let your domains transfer to your new host. You just have to change the DNS settings and don't have to hassle with your host provider.

That's about it, good luck with your move! And if you have further tips, let us know!