Trouble with I watch five in music. Can I download music from iTunes or cannot only listen to music using

To download songs:

Add music to Apple Watch - Apple Support

To play the songs:

Play music on Apple Watch - Apple Support

Note that the link says:

"After you connect Apple Watch to Bluetooth headphones or speakers, open the Music app  on your Apple Watch, then ..."

and that link says:

'"Tip: If you have AirPods that you set up with your iPhone, they’re ready to use with your Apple Watch—just press play."

You can play music via your Apple Watch. But you must use Bluetooth headphones or Bluetooth speakers. You cannot play music via the Apple Watch speakers, as the tiny battery will not last very long if this were allowed.

You do Not need to use Apple AirPods. I've successfully used 4 different pairs of Bluetooth headphones paired to my Apple Watch. Sony Noise Canceling headphones, AirPods, and two different 50+ year old Bluetooth headphones. The most difficult part is finding the instructions on how you tell the headphones to enter "Discovery" mode. Once you figure that out (with lots of Google searching), the rest is easy.

Listen to music, podcasts, and audiobooks on your Apple Watch

https://support.apple.com/en-us/HT204691

Add music to Apple Watch

https://support.apple.com/guide/watch/add-music-apd483798d11/watchos

Play music on Apple Watch

https://support.apple.com/guide/watch/play-music-apd70768b20b/watchos

Be aware, that Spotify (a popular music streaming service does not support the Apple Watch, beyond having the Apple Watch act as a remote control for the Spotify app on your iPhone.

Apple Music and Pandora both offer music streaming to the Apple Watch.

And you can transfer your own music files to the Apple Watch and play them without needing any network access. Pandora also offers the ability to tranfer off-line playlists to the Apple Watch for playing without needing network access.