The install is failing because you are missing this folder. Seems like npm thinks that module is already in your .staging folder but it’s not. These types of errors are super common, sometimes npm messes up it’s cache. A quick google of: “npm ERR! enoent ENOENT: no such file or directory, rename '/usr/local/lib/node_modules/.staging” lead to a bunch of results, one of which will most likely fix your issue.
In fact, if you search these exact forums…someone has already had literally your exact same word for word problem: [solved] First time Ionic 2 installation problem
Have you tried that fix already?