SCP or SFTP are your friends in this scenario.
scp command works similarly to
cp, but it copies files over an ssh connection.
eli@oldserver:~$ scp -r /folder/with/important/stuff/ email@example.com:/home/eli/stuff
For more information and flags I refer you to the man page of scp.
Your other option, SFTP is a protocol that is implemented in many file browsers. This approach will allow you to use a graphical interface.
- File browsers like Nautilus and Dolphin on desktop Linux distributions have this built in. Just use
sftp://firstname.lastname@example.org in the address bar.
- On Windows there is WinSCP.
- For OSX there is Fugu.
These SFTP tools will allow you to just drag and drop files on or off servers running ssh.