I am using VueJs with Laravel and am using the Laravel's default installation of VueJs with their Laravel-Mix.
I want to use Babel, Eslint and Vue-Router all of which don't seem to come with Laravel's default installation?
How can I use the Vue Cli to handle all of this with Laravel or do I need to pull everything in separately, something that the Vue Cli was built for?
The Vue CLI is meant for bootstrapping SPA's from scratch where the main intent is a Vue.js powered website.
You are using Laravel here which always has everything bootstrapped so I guess it would be easiest to include it yourself. From my understanding Babel is already included.
If you want to also use ESLint and vue-router you can look at the Github repositories from the Vue.js CLI on how to set it up:
https://github.com/vuejs-templates/webpack
It is not that hard to set up.