This is not something AE does out of the box, as our role is more to allow easy social sign-on and tracking and then hand over to you to make use of the information as you wish. That being said, AEJS is set up to make it easy for you to build a custom application that will follow playlists for you after AE signup.
The first step is to log into your AE Dashboard, click AE Connect and add a Spotify application. This will allow your users to sign in with Spotify, which will let you track their behaviour.
Here's a guide to creating your Spotify app:
Once this is done, you'll need to add the JS framework to your site, which will allow you to add Spotify functionality in conjunction with AE to your website.
Check out our setup guide here:
Finally, adding playlist functionality or Spotify follows isn't something AE does out of the box -- you'd need to set this up using Spotify's web API.
You could then hook this into AE's login callbacks on your site. For example, once the user has logged in using AE and Spotify on your site, you could trigger AE's onLogin event and make the second call to Spotify's API to trigger the playlist follow.
Here's some info on AE's callback functions that you can use on your site once the framework is set up: