Moving isn't too hard. What you'll want to do is copy all of your files of course.. and then your database. Change configuration files to point to your new paths and database settings.
This is a neat trick though.. once you get everything moved you may not have a way to test if it's working.. The final step is to repoint the DNS to the new server's IP but you dont' want to do that unless you know your new site is up and running good.
The easy way to fix this is to hack your computers HOSTS file. Search the net for specific instructions.. but you can override DNS for your computer and hit your new server.
Once you've verified it's all running you can modify your DNS. At this point I'd put your old cart in maintenance mode for 24 hours or so to prevent anyone from making orders and changes there. Hopefully they will be redirected to the new site and nothing will be noticed.. but it takes 24-48 hours for DNS to propagate.
You can be all snazzy and set up an external database connection so that you don't have any downtime.. I did that with my move awhile back.. but the external database connection on the old server was slow as heck. Maintenance mode might be the better choice from a usability standpoint.