apt basics:

apt-get install ${package}      : install a package
dpkg --list                     : list installed packages
apt-get remove ${package}       : remove a package
apt-cache search ${pattern}     : find packages matching pattern
apt-cache show ${package}       : show info about a package
sudo apt-get install apt-file   : apt-file needs to be installed first
apt-file show ${package}        : shows files in a package
apt-get upgrade                 : upgrade all installed packages

Note: apt-file might need to be installed first (sudo apt-get install apt-file)

dpkg basics:

dpkg -i foo.deb      : install a package
dpkg -l | grep foo   : list installed packages, show those matching foo
dpkg -r foo-ver      : remove pacakge

rpm basics:

rpm -ihv *.rpm         : install a package
rpm -Uhv *.rpm         : upgrade a package
rpm -qa | grep -i name : list all packages matching a name
rpm -ql $package-name  : list files in a given pacakge
rpm -q --whatrequires mysql-server : list packages dependent on a given package
rpm --recompile $package-name      : build and install a source package