How do I confirm iCloud contacts backup and transfer to a new iPhone?

If you synced contacts with iCloud you can use any internet browser and log into www.icloud.com and view the synced contacts there.

If you didn't sync the contacts, they are included in iCloud Backup. See: What does iCloud back up? - Apple Support

You cannot view the contents of an iCloud Backup but you can view the date and size of the backup by going to Settings > [username] > iCloud > Storage > Backups > [device name].

"seeing backed up contacts: I just backed up my iphone and set up my account. How do I know they have been saved and can be used to populate a new phone?"

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Use QuickStart for the Transfer:

As for transferring to a new iPhone, this is the link to go by: Use Quick Start to transfer data to a new iPhone or iPad - Apple Support. Contacts will be transferred, if you select that.