apt vs apt-get:

apt-get and apt-cache's most commonly used commands are available in apt.

Update vs Upgrade(apt):

You should first run update, then upgrade. Neither of them automatically runs the other.

apt-get update updates the list of available packages and their versions, but it does not install or upgrade any packages. apt-get upgrade actually installs newer versions of the packages you have. After updating the lists, the package manager knows about available updates for the software you have installed. This is why you first want to update. Additionally, you can use apt-get update && apt-get upgrade to do both steps after each other.