Set up on the new server the identical site and test it works. You can usually preview the site via the ip or change your hostsfile on your computer so when you enter the web address it takes you to your new ip (but the rest of the world see's your old site still).
Get it all up and stable, check everything. All you need to do then is change the dns to point to the new server.
If you have a systems admin you can get them to change the ttl a few days before to a smaller number so that the dns transition is picked up more quickly around the world.
Some important rules:
- Don't do it on a Friday as you might need to roll back something if it goes wrong and you may need support staff who could be unavailable on a weekend.
- Give yourself plenty of time. More than you think
- Don't forget to update any email dns settings etc
- Make sure everyone is aware email might be delayed while dns is transitioning
- check your dns transition across the globe here http://www.whatsmydns.net/
- Don't switch off you old hosting for at least a few days maybe a week. Some websites may still be reading your old ip address and you never know you might need to do an urgent roll back to the old server if something nasty happens.
If you need a systems admin who can do this all for you I can heartidly recommend mine.