To update Linux Mint from the command line, use these two commands.
sudo apt-get update
This, I think, refreshes your system's list of what updates are available.
Then
sudo-apt-get upgrade
This actually replaces what's on your system with newer versions. Mint (I think) does dependency checking automatically, so that if the new software needs new parts underlying it, Mint will install those automatically as well.
The second command will present you with a list of what's about to be replaced. You'll have to OK the replacement before it starts.
Of course, there is probably a graphical way to update, too.