Follow these steps to ensure ZERO downtime and to ensure a seamless move to your new host.
1.) DO NOT CHANGE YOUR NAMESERVERS JUST YET, NOR DELETE YOUR HOSTING ACCOUNT WITH YOUR CURRENT WEB HOST!
2.) Establish a hosting account with your new host. You'll be able to access your new account via the IP address of the server your new account is on. Upload all of your files to the new server.. Verify that everything is working on your NEW site (web pages, forums, links, etc.).
3.) Once verified, go to your domain registrar and change your "NameServers" to those that were provided by your new host.
4.) Your site will be updated WORLDWIDE within 24-72 hours. Note: Although you may be able to see your site is on your new server within possibly 12 hours, please allow a complete 72 hour cycle before you delete your old hosting account completely.
6.) Enjoy your new hosting!