If you want to save your progress, be sure to check the "Profile" tab. It is located in the bottom part of the in-game lobby.

 

Here, you can sign in using Facebook or Apple ID (if you are playing on an iOS device). After that, you can log in to your account on any device by simply signing in to the very same Facebook/Apple ID you originally logged in with.

 

Once you do so, your profile picture and name will be uploaded from your Facebook/Apple ID account.

 

You can still change your avatar by clicking on it and selecting the suitable option. You can also change your nickname by clicking on the pencil button next to the field with your current nickname.