Do I also need Apple Music with this new subscription

They are two separate services that provide different benefits. iCloud storage provides you with storage and Apple Music allows you to listen to streaming music. See: Apple Music - Apple

Apple offers bundles where you could combine those two services with others but there would be a higher charge. See: Apple One - Apple

If you decide to cancel Apple Music:

There is an Apple article that explains how to cancel subscriptions. Click on the following link: If you want to cancel a subscription from Apple - Apple Support

or use an internet browser to cancel as described in How to cancel Apple Music - Apple Support

See if Apple One may be cheaper for you.

Bundle Apple subscriptions with Apple One - Apple Support