How can I get my SIM card to work in a new iPhone that has a different carrier?

If it was a much longer message that also told you this was not a hardware error then your new iPhone is locked to a particular carrier. You can find out who in Settings > General > About. You will either have to switch to that carrier or see if you qualify to unlock it.

If your phone is carrier locked, you will need a sim card from them or ask them to unlock your phone.

If the new phone is unlocked then you put the old SIM in the new iPhone and then activate it.