@dannymcc has a good suggestion here. Speaking from experience, the following is a suitable and reasonably fault tolerant way to move your site:
1) backup the setups (files, settings, databases etc.) from your old data-center
2) move a copy to your new hosting. Setup your web server on the new server as if it is already using the live domain
3) hard code the hosts file on your local machine so that your domain points to your new server's IP (see http://www.techrepublic.com/article/keeping-your-sanity-with-etchosts/5033406 for Linux and entry about Static Clients on this page for Windows http://support.microsoft.com/kb/309642)
4) once you are sure that you're looking at the new server for your domain on your local machine - that is, you are forcing your local computer to resolve the domain to your new server - test that the site is working properly. HINT - hard code a piece of text into the page on the new server like "This is the new server" so you can be absolutely certain you are looking at the correct server
5) once you're certain that you're looking at the new server and everything is working correctly, you can start editing the zone entries (specifically the A-record) for your domain. You can use the NSLOOKUP command to find out where your nameservers are. Often they are bundled with your domain registration, but for advanced setups, this may not be the case. More often than not you will get some sort of web based editor for changing these - if you need to update it over a command line for something like BIND, I recommend you get someone experienced to do it
6) edit your website A-record (or other records as desired) so that they point at the new server. It will take some time for the DNS to resolve for everyone, and until you're sure that everyone is being redirected, I'd suggest leaving both copies up (you could check the server logs of the old webserver to see if it is still receiving significant traffic). It's difficult to say how long this will take, but my guess would be between 8-36 hours.