Go to Settings » Synchronization » Synchronize notes. You will now be asked to grant Notes limited access to your Dropbox or Google Drive account. After that you can synchronize by swiping down on the list of notes.
If you also enable Automatic synchronization, the app will automatically sync after creating/editing a note.
Using Notes on a second/third/nth device
Once you have synchronized on one device you can simply repeat the process on any other device you would like to use Notes. You can also use the web version to access your notes.