How to make Yakuake and any other program start at OS Startup

You can add an item in ~/.config/autostart,
e.g Open ~/.config/autostart/myscript.desktop, and paste the following contents
[Desktop Entry]
Name=MyScript
GenericName=A descriptive name
Comment=Some description about your script
Exec=/path/to/my/script.sh
Terminal=false
Type=Application
X
-GNOME-Autostart-enabled=true
And make sure the script is executable.