In order to listen to music in the pool, you will need a paid subscription to an Apple Music account that allows offline listening: Either the $9.99/mo Individual, or one of the Family or Student accounts will work. A free membership will not be able to play offline while swimming. This link will take you to Apple Music’s website for learning more about the different subscription tiers.
Apple Music is not the same as iTunes. If you have an Apple Music subscription, you can use the Apple Music App to listen to your purchased iTunes music.
However, you don't need the Apple Music App to listen to iTunes music on your Delphin. To download your iTunes songs, you can plug your Delphin into a computer and Drag and Drop files into the Music folder on the Delphin. You can then use one of these MP3 Apps (Pi, BlackPlayer, VLC) to listen to your AAC files from iTunes. Files purchased before 2009 will be protected and therefore not usable on the Delphin.
There is a workaround for older files, however, it requires you to delete your purchased files and then redownload them from Apple. This process can be tricky. Feel free to contact Apple for more information on how to do this properly.