I only want to back up the address book

I suppose that you could export contacts in an e-mail to yourself, then wipe the phone and set it up from your old one, then import the contacts from e-mail.

Export contacts on iPhone - Apple Support

Import contacts on iPhone - Apple Support

What to do before you sell, give away, or trade in your iPhone or iPad - Apple Support

Use Quick Start to transfer data to a new iPhone or iPad - Apple Support

As far as malware, unless your father's phone is jailbroken, it is unlikely that it contains any malware. Be aware, though, that many social media apps and store loyalty apps (whether installed by your father or by you) may try tracking you relentlessly, for purposes such as selling your profile to advertisers.

Control what you share on iPhone - Apple Support

mamasnowwolf wrote:

1. My dad got an iPhone 15, but no longer needs it so he gave it to me. I have an iPhone 12. I want to keep some of my dad's phone numbers, but nothing else on the phone will stay I want to make it my own. So I have some big questions that I can't seem to find answers to.

How do I save just the phone numbers? I don't want to save apps or passwords just the phone book.

2. My phone company said that if I do a factory reset on the phone, the line will be down temporarily (a day or 2) so they can set up the phone with their info. I can't really be down that long is there a way to make sure there isn't any malware on the phone?

3. If I just sign out of my dad's iCloud account and sign into mine, will that allow me to install the phone book from my phone as well as passwords to apps and such? Or is it going to be more complicated?

Thanks in advance.

Add the contacts your want from your dad's phone to your phone.

Then wipe his phone What to do before you sell, give away, trade in, or recycle your Mac - Apple Support

Once done you can do this Transfer data from your previous iOS or iPadOS device to your new iPhone or iPad - Apple Support

Not sure where you're getting malware from there are no know malware/spyware/virus/hack for a non jailbroken phone.