Flying to Scotland from the US, where do we check in?

If you booked both flights on the same ticket, you'll normally do all your checking in at the first airport.

If you booked the flights separately, then you will normally check in twice, once for each flight. For the second flight, the airline won't pay attention to how you're arriving at the airport, whether by plane, train, taxi, on foot, or otherwise. All they care about is that you're there on time.

Your question is a little puzzling because checking in for an international flight is not much different from checking in for a domestic flight. The only difference is that the airline will verify that you have adequate documents to pass through the immigration checkpoint at your destination. For a US citizen going to Scotland, this means that they will check that your passport is valid.

Even if you check in for both legs at the first airport, you might still have to show your passport again at the second airport. Sometimes there's a detailed immigration document check at the gate before you board; sometimes they just verify that your passport and boarding pass match when when you present the boarding pass to get on the plane.