Where in Settings can I see if my carrier has unlocked my iPhone?

Go to Settings > General and tap About.

Under Carrier Lock, you should see a message that says, "No SIM restrictions." If you don't see that message, contact your carrier.

Source.

If you got the phone through your carrier, your carrier is not going to automatically unlock your phone, even if the phone is paid for. You have to contact them to ask them to unlock the phone.

Things might be a lot easier for you if your carrier offers an "international" option.

We've been to 8+ European countries in the last few years, and it is really nice to power on your phone and have it automatically connected to the local carrier ready to go wherever you go.

Sure, it cost a bit more, but the convenience is more than worth it, having tried the "find a carrier in every country" fiasco before.