which sim card, and how to use?

Yes, you can get a SIM card at the airport. The people at the store can give you information about which card might work best for you, both as far as coverage and the other things you need (data, calling, etc.).. You don't want to keep switching SIMs, though. Every time you switch SIMs you also switch phone numbers, and there is no reason to do it. What might work best for you is a pay-as-you-go card that you can add time to as you need it. The people at the store can also show you how to make local calls, and dial back to the US if you need to.

nancy - thank you. so is that a pay as you go sim card? and it will give me a different phone number than the one that i have with AT&T - that makes sense. so the pay as you go card has its own phone number? i am supposed to give all the places we are staying, a contact phone number. i guess giving them the phone number i have now won't be any use.

jbsamwise, Correct. The SIM card is equivalent to the the socket you plug your home phone into. It decides who bills you for the calls and what your phone number is. New SIM card = new phone number, put back your old SIM card and you get your old phone number again.
Any SIM card you buy in the UK will have a UK phone number. So you dial a UK number same as from a landline, for example 01234 678901. And you dial a foreign number: 00{country code}{area code}{number}. For example, for the US: 001aaannnnnnn.
There are some phones that can take two SIM cards, but that is getting complicated.

jb,

If you're going to be in Edinburgh for a few days, you'll have a greater range of choice in accessing cell shops there. The staff at your hotel can provide information on the nearest one. It would be a good idea to verify that the phone is indeed unlocked, as it won't work with other SIM cards unless it is.

The staff at the cell phone shop will be able to provide some information on how to use the phone. You'll likely have several different plans to choose from, so you'll have to give some thought on how much you'll be using voice, text or data. You'll also need to be clear on the method for "topping up" the card when your time is depleted.

This chart may help with the method for making calls in Europe.....

https://www.ricksteves.com/travel-tips/phones-tech/call-uk-europe

i am sure it's an unlocked phone, as i have a pay as you go sort of set up with AT&T, and he said that that makes it an unlocked phone. that if i had any other kind of plan, it would be locked into AT&T. we will only have the day we arrive, a full day after that, then leave at check out time for the highlands. so not much time for going to the edinburgh phone stores. better just buy one at the airport. (on the other end of our trip we will have more time in edinburgh). Thank you for the information on how to make the calls. will make sure i have that with me.

back from an absolutely incredible month in scotland. thank you to everyone. i ended up putting an 02 sim card in for the month and it worked well. there were occasional areas where it wouldn't work, but nothing that caused us difficulty, and those were areas that I would imagine no other card/phone would have worked either. again, a thank you to everyone for helping me to understand and use the sim card!

I looked at the list of shops located in Edinburgh Airport, but none seemed likely to carry SIM cards. I realize there are some Car Warehouses a few miles from the airport, but for a host of reasons, I really need to get the SIM cards for all of our phones at the airport. Is there a specific store or kiosk in EDI that sells them?

Thank you.

I have never been to EDI, but I expect that the WH Smith (one each before and after security) would have some PAYG SIM cards near the tills.

Thanks for the update!

My daughter who recently finished her studies in Edinburgh, purchased & used the O2 sim card for her unlocked iphone, but most of her friends had the
"Three Co" Sim card: http://www.three.co.uk/Store/SIM