It would be good practice to first set up the grid account, but not move anything.
Second, change youe name servers, but have the dns on your grid server pointing to the OLD ip address, with a very low TTL. Low as in 5 minutes (300 seconds).
Wait a few days for the name server change to propagate.
While you're waiting, you could go ahead and copy over most of the files and save some time later.
Once you're sure the name server change has propagated, may take a couple of days, then late at night, turn off your current site.
SSH into your account, and make a tar of your blogs.dir.
SSH into your new account, cd to your wp-content dir, and wget the tar you made on your other host. (Host to host is (or should be) much faster than host to you, and up to new host.
Then do a dump of your database, you may want to segment it a bit, and then import it into your new host.
Next modify your local hosts file to point your domain to the new ip address. That way you can test it to make sure it works.
Once verified, change the IP in your DNS back to pointing to your grid server ip.
Not supported, warranted, guaranteed, etc. Advice is at your own risk and for illustrative purposes only.
After that, upgrade your site once it's been rolling for a few days on the new host.