pacman -Sy #sync the database
pacman -Syy #sync the database even if it's up to date
pacman -Su #update installed package
pacman -Syu #sync the database and then update
pacman -Syyu #force sync the database and then update //<i>recommended before any package installation</i>
pacman -S <package> #install package
pacman -Sw <package> #download package without installation
pacman -U /package_path/package.pkg.tar.xz #install downloaded or local package //Usually package_path = /var/cache/pacman
pacman -Sg <group> #search package installed in a group
pacman -R <package> #remove package
pacman -Rs <package> #remove package with dependecies that are not being used by another package
pacman -Rsc <package> #remove package with dependecies even if needed by another package //Common sense is advised
pacman -Rn <package> #remove package and its configuration files
pacman -Rsn <package> #remove package with (self-needed) dependencies and its configuration files
pacman -Sc #leaves packages in cache only for packages currently installed //No possible to downgrade after this
pacman -Scc #clean cache completely and remove all packages //No possible to downgrade after this
paccache -r #remove old package cache files but the latest three versions of package either installed or not
paccache -rk n #keep n latest versions of a package and remove the rest
paccache -ruk0 #remove all cached versions of uninstalled packages
pacman -Ss <package> #provide description of searched package and associated package
pacman -Si <package> #provide detailed summary of package
pacman -Qs <package> #provide description of previously installed package
pacman -Q #get a full list of installed packages
For more detailed info, pacman-ArchWiki and Manjaro's pacman tips.