If you know you are going to be changing servers, it's best to set the TTL (Time To Live) option on the DNS records to a low value such as 5 minutes rather than the default which is between 4 and 72 hours. The TTL is a value which tells computers how frequently the domain name may change. This is used to help speed up web page delivery. Think of it like a phone book. If every time you had to open the phone book to find a phone number, your dialing time would take longer. So instead you write it down on a piece of paper (which magically disapears every 4 hours). :)
So by changing the TTL to 5 minutes you're telling computers to come back and check the number as it may have changed. Once you've successfully moved to the new server, set the time back at the 4-72 hour range so as to speed up your users visits to your website.
To change the TTL you would contact the place where your nameservers are pointed to and ask them to change the TTL.
Here at Hands-on Web Hosting, the TTLs are set at 4 hours and then dropped to 5 minutes if we are moving a site from one server to another. If your DNS is not pointing to Hands-on, it's recommended to contact your DNS Provider and change the DNS prior to changing servers so as to get a smooth transition and not leave any users visiting a blank site (on the old server).