Do I need to back up my phone and how do I do it?

It is a good idea to have data synced with iCloud and to have a backup. Phones can be lost, stolen or damaged and without that your data will be lost also.

Some data can by synced. One advantage of syncing over backup is that you can view the data in an internet browser at iCloud.com. to sync Photos, for example, see instructions in Set up and use iCloud Photos - Apple Support

Other sync options (e.g., Contacts, Calendars, Notes, etc) are at Settings > [username] > iCloud.

Data that can't be synced (including data that you choose not to sync) is included in iCloud Backup or backup to a computer. See instructions in How to back up your iPhone, iPad, and iPod touch - Apple Support

Also see: Backup methods for iPhone, iPad, and iPod touch - Apple Support