Use FlashFXP, this ftp client supports server-to-server transfer.
This means you don't have to download your current files and upload it to your newer server.
It means that the downloading content of content on your current server will be uploading to your newer server.
1) Move the data from your current ftp server to your newer ftp server.
2) Backup your current MySql (or other) database and restore it in your new server.
If you have a non-standard website (where these rule perhaps aren't possible)
Please provide us with more information about your server.
