installing contacts from my old sim card to my new iPhone

iPhones don't save contacts to the SIM. They may just be on the phone itself, or they may be synced with iCloud or another email account.

On the old phone, go to Settings>Contacts, and see what's listed for Default Account. that's where your contacts are stored.

~Lyssa

What did the old SIM card come out of?

iPhones don't store any data at all, including contacts, on the SIM.