Nowadays, the Spotify app is available for Android devices also. You can directly download it on the Google Store. After that, you can play Spotify playlists online via the Shuffle Play mode. With a free Spotify Premium, it has many limitations to listen to playlists.
If you'd like to get a better listening experience, the best way is to upgrade to Spotify Premium.
If there is something wrong happened, and Spotify not working well via the app, you can try to use AudFree Spotify Music Converter to download Spotify songs as local files after that you can still listen to Spotify playlist on Android via any music player instead of using the Spotify app.