I don’t think there’s a link that explains iCloud completely.
To make sure there’s a most recent backup, go to Settings - tap your information, go to iCloud - iCloud Backup - "Back Up Now". Right below that, you should see the time of the last backup. To backup manually, tap "Back Up Now".
To get your contacts to a new iPhone, you don’t need a backup.
Just set up the email address which stores your contacts on your other device, enable Contacts, and they will be synced to the iPhone.
When you activate an iPhone, it will ask you to login to your iCloud account. When you’ve done this, go to Settings - Mail - Accounts, select your iCloud account, and make sure Contacts is enabled.
To set up an email account other than iCloud, go to Settings - Mail - Accounts - Add Account, and set up the email account.