The codex entry for Moving WordPress doesn't make sense to possibly ignorant person. Allow me to elaborate.
If You Don't Care If Your Old Blog Works
1-Download a copy of the main wordpress files from your OLD blog to your hard drive and edit wp-config.php to suit the new server.
2-Go back to your OLD blog and go to Administration > Settings > General and change the url (both of them) to that of your new site.
3-Again, download your database (but keep the old one just in case), upload this new database and the copy of the wordpress core files with the edited wp-config.php to your new server. Thats it!
The thing is once you change the url of the site you can no longer access the dashboard (of the old site) and hence you can no longer download the database that has the new urls on it.
What am I missing?