Southern Spain - arrival airport/departure airpirt

In March I went to those 3 cities + Madrid. I took American Airlines to and from Madrid. I took the train from Madrid to Seville, bus to Granada, train to Cordoba, train to Madrid. When I bought my plane tickets, I wrongly believed I had to only take flights on a United States-based airline. I could have taken TAP Portugal to Seville, switching planes in Lisbon instead of Madrid. When I first searched for tickets, it was possible to fly to or from Granada. I don't know whether it is currently possible to fly to or from Granada. It did not occur to me to fly to Malaga and then take the train to Granada or Seville. See if it is possible to fly to Seville and fly back from Malaga or fly to Malaga and leave from Seville. If going round trip, fly to and from Seville, make Seville your last city. Without spending a night in Seville first, after you leave the airport in Seville, take the train to Granada or Cordoba. Which one you go to first depends on which day of the week you arrive on and what days of the week you want to be where and which of the sights will closed on Mondays or Tuesdays. You need to write a day-by-day itinerary. Before your final draft, you need to experiment with Cordoba and Granada as your first city, paying close attention to when each sight you want to see will be open, both which days of the week and the times of the day the sights are open.

If you fly American they can codeshare you onto a flight to either Granada or Seville from Madrid on Iberia, or fly Iberia the whole way.

Malaga is not particularly useful for you. You don’t want to be there and I think it’s only a bus to Granada.

Your struggle is probably because Malaga is not a big international destination and Sevilla's airport is much smaller. If you can fly non-stop to Madrid, that's probably your best bet. If you land in the morning, or even early afternoon, you can get to Cordoba by train from the airport with a change in Madrid. Flying to Sevilla via Madrid will probably take longer. You would need to spend your last night in Madrid.

Your other option is to fly via a major European airport to Sevilla. Malaga's not much use as Tom points out. I'd start by going to the wiki page for Sevilla's airport where you'll find a table of airlines and the cities they fly to from Sevilla.

Madrid may very well end up being the most economical solution, but remember you'll have to pay for train tickets and possibly spend your last night in Madrid.
I have flown in/out of Sevilla, Jerez de la Frontera, and Malaga--for not that much more than I would have spent for MAD flights. Check all of your "multi-city" options.

I really appreciate everyone's replies! Thank you. What still confuses me is I don't want to back track. So if I land In Seville and spend time there, then head off to Cordoba, and finally Granada where does the trip end as it relates to flying back to the US? I guess I can take a train from Granada to Madrid (3.5 hours) and fly home from there.

You'll have to check the flights. I recall Grenada>>Madrid>>USA not working out very well timewise, but you could check the itinerary into Granada and home from Seville. Note that the high speed train that connects all the cities negates some of the "avoid backtracking since it wastes time" issue. The layovers in Madrid were pretty long on the Granada flights I checked (I ended up not going this time).

You have to do the legwork and compare the various possibilities.
Many land in Madrid and then hop on the high speed train to Cordoba, that worked well for me. You could do Granada next and possibly fly out of Sevilla.