Don't install Node.js directly from apt
—it's hideously out of date, and most of the features you're expecting to have won't work. The version in the repositories at the minute is v0.10.29v0.10.29, whenwhile the current release as of writing is v7.10.0v7.10.0.
Instead, follow the advice given to install Node (and npm) from their website:
curl -sL https://deb.nodesource.com/setup_7.x | sudo -E bash -
sudo apt-get install -y nodejs
Note that this will only work for Pis with an ARMv7 (or greater) processor—the Pi Zero and the original Model B won't cut it, as noted here.
After completing the installation, both Node.js and npm will be installed using the latest (or a very recent) version.