There are two main Tools you may use:

1) The Developer Tools in Chrome (see below)

2) The Vue Developer Tools (https://github.com/vuejs/vue-devtools)

When working with a Project created by the CLI (using Webpack!), you can easily debug your running Application by opening your Developer Tools (see below in Chrome) and going to "Sources" there. 

You should see a "Webpack" folder which you may open to see all your files included in the project (see image below, in orange box).

You can open any file there and set breakpoints etc. to debug your running application.