Spain in 10 days. Planning flights.

See if there's a flight to Granada. From Granada you can take the train to Madrid and then train to Barcelona, then fly back to Pennsylvania from Barcelona. Otherwise look for a flight to Madrid, trains to Granada, bus, train, or plane to Barcelona, leave from Granada. It is logistically better to do an open jaw itinerary, but choice depends on price versus trip durations versus enough but not too much layover time versus time the planes leave.

Time is your most valuable commodity on a vacation so we have used open jaw flights to make the best use of our time. Yes, will cost more but consider this:
Flight via connection in Madrid to Granada keeps you on same efficient mode of transportation without need to leave airport with luggage to connect with a train. We have used Iberia to fly in this manner to a number of different cities to commence our vacation (Granada, Malaga and Seville). Whatever the itinerary you choose highly recommend open jaw flights.

Open jaw is generally a better idea because it saves time. Fares are usually comparable to a round trip to either city. Keep in mind that backtracking costs money as well as time. In your case, Barcelona and Madrid are both big destinations with lots of flights and might work for you. Not so sure about Granada, you might be better off taking the train from Madrid and back. But as a previous poster noted, there's efficiency in checking bags through with a short airport layover instead of transferring from airport to train. Your time on vacation, especially this limited, is worth some money. I haven't done any research for you here, but it's easy enough to do yourself using the "multi-city" option on a search engine like Kayak or Google Flights.

Fly into Madrid and open jaw out of Barcelona.

We took a coach bus from Madrid down to Granada. I think that they have a train now that makes that route.

We stayed in an apartment right below Alhambra in a residential neighborhood. A comparable place on Booking.com is Acogedor y silencioso Apto para turismo y relax Catalina 2. Apartments in the area are very inexpensive and you can get close to the local community by staying in one there.

We caught a Vueling Air flight over to Barcelona--a 1 hour flight. Spain is larger than most people realize. They're a budget airline so the flight was very inexpensive.

If it’s 10 days wheels up/down that means you lose a day going over, leaving nine. The last day is travel, so that leaves eight. For me, I’d split the time between Madrid and Barcelona and skip Granada if it’s your first trip to Spain. Fly into one and out of the other.

Thanks so much for all of the advice! We just booked flights to fly in to Madrid and out of Barcelona. May 1 to May 11. Added an extra day on the ground.