apt-get install <package> Downloads <package> and all of its dependencies, and installs or upgrades them.
apt-get remove [--purge] <package> Removes <package> and any packages that depend on it. --purge specifies that packages should be purged.
apt-get update Updates packages listings from the repo, should be run at least once a week.
apt-get upgrade Upgrades all currently installed packages with those updates available from the repo. should be run once a week.
apt-get dist-upgrade [-u] Similar to apt-get upgrade, except that dist-upgrade will install or remove packages to satisfy dependencies.
apt-cache search <pattern> Searches packages and descriptions for <pattern>.
apt-cache show <package> Shows the full description of <package>.
apt-cache showpkg <package> Shows a lot more detail about <package>, and its relationships to other packages.
man apt Will give you more info on these commands as well as many that are in less common usage.