(A) Few days before moving your site to the new server, implement cloudflare on your site. (i.e if you are not already using it)
Its free and the process is pretty straightforward.
Switch to Cloudflare nameservers and wait for 2-3 days for DNS propagation.
(B) Move your site to new server.
database driven sites (wordpress, vbulletin) should be put into read only mode before moving.
(C) Update cloudflare DNS settings to reflect the IP of your new server.
After that if you do not wish to use cloudflare just update nameservers at your domain registrants but DO NOT delete domain from cloudflare