Here are detailed instructions combining getting rid of all MySQL from your Mac then installing it The Brew Way as Sedorner wrote above:

Remove MySQL completely per The Tech Lab

• ps -ax | grep mysql
• stop and kill any MySQL processes
• sudo rm /usr/local/mysql
• sudo rm -rf /usr/local/var/mysql
• sudo rm -rf /usr/local/mysql*
• sudo rm ~/Library/LaunchAgents/homebrew.mxcl.mysql.plist
• sudo rm -rf /Library/StartupItems/MySQLCOM
• sudo rm -rf /Library/PreferencePanes/My*
• launchctl unload -w ~/Library/LaunchAgents/homebrew.mxcl.mysql.plist
• edit /etc/hostconfig and remove the line MYSQLCOM=-YES-
• rm -rf ~/Library/PreferencePanes/My*
• sudo rm -rf /Library/Receipts/mysql*
• sudo rm -rf /Library/Receipts/MySQL*
• sudo rm -rf /private/var/db/receipts/*mysql*
• sudo rm -rf /tmp/mysql*
• try to run mysql, it shouldn't work

Brew install MySQL per user Sedorner from this StackOverflow answer

• brew doctor and fix any errors

• brew remove mysql

• brew cleanup

• brew update

• brew install mysql

• unset TMPDIR

    mysql_install_db --verbose --user=`whoami` --basedir="$(brew --prefix mysql)" --datadir=/usr/local/var/mysql --tmpdir=/tmp # whoami is executed inline
  

• mysql.server start

• run the commands Brew suggests, add MySQL to launchctl so it automatically launches at startup

mysql should now work and be running all the time as expected