Here's a general checklist to follow to ensure that you have the smoothest move possible, and hopefully with no down time.
Step 1: Get all of your files local.
Using a basic FTP program or whatever development tool you may be using, download all current files used in your web site including graphics, html files, and anything else that may be used in your site to your local hard drive (if you don't already have it). Most likely you already know this - but when you are copying down your files you want to keep the directory structure exactly as it is on your web server. If you built the site then most likely you already have all of this.
Step 2: Get everything loaded to your new server.
Before making the domain name change, go ahead and upload all of your site files to the WHR server. Although they won't really work yet, go ahead and set up all of the email accounts that are used on your domain as well.
Step 3: Initiate the domain name record change.
Either through Network Solutions or whatever registrar you used to register your domain, initiate the name change. Technically speaking the only thing that really need to be changed in your record is the name server information. You should receive this information included in your "welcome email" from us after signing up for an account with us. If you need any help or additional information, just submit a support ticket. It is also a good idea to update the technical contact on your domain name record with our contact information, although that is not required.
Step 4: Monitor for the domain name change.
Depending on who you used to register your domain through, you should get one or more emails confirming the domain name change. Once the change has been initiated it will typically take 24 - 48 hours for the entire world to see the change. This period is called propagation and is simply the time it takes for all the DNS servers around the globe to "catch up" and take note of your domains new location. Once propagation has completed its course you are free to safely cancel service with your previous host.