if you have a control panel on your server, it's much easier. If you don't, you will need to do manually. In that case, it depends on which OS you are using, databases etc, then there are different commands to do it
you'll want to login to your control panel for that domain, create a full backup, usually in the form of tar.gz and then you'll want to provide him the backup. He can create an FTP account for you to do this and then his host can restore that backup, etc..
DO you have Cpanel control panel ? If You use CPANEL : Log in to Cpanle --> Backup ---> Generate A full Backup ---> Select Remote Backup Passive_Mod ---> write remote IP and U + P Then You can get this backup from second server. Thanks
Manually transffering the files are not the best of the ideas. In case the control panel is plesk, you may transfer the data using psa dump and in case its a cpanel, server through the whm or the pkgaccnt. Post here whether you have any control panel installed on the servers you are planning to transfer.
Download all files to your PC using the FTP client. After that Upload the files to the new host. Or send files to him/her, he/she can upload in their host and Done !
ftp download/upload to your PC is not always good. Yesterday, I did the migration job for one user here in DP, and it took 5 hours for completing. When data size is GB, it's better to migrate directly between servers, not to/from your PC