This is the problem with your $PATH:
/usr/local//usr/local/mysql/bin/private/var/mysql/private/var/mysql/bin.
$PATH is where the shell searches for command files. Folders to search in need to be separated with a colon. And so you want /usr/local/mysql/bin/ in your path but instead it searches in /usr/local//usr/local/mysql/bin/private/var/mysql/private/var/mysql/bin, which probably doesn’t exist.
Instead you want ${PATH}:/usr/local/mysql/bin.
So do export PATH=${PATH}:/usr/local/mysql/bin.
If you want this to be run every time you open terminal put it in the file .bash_profile, which is run when Terminal opens.