9 days in southern Spain

If you have not purchased airfare then please examine flying to Granada, instead of arriving in Madrid.
If you have purchased airfare then suggest you travel from Madrid to Granada and start your trip.
Depart Granada and go to Seville via train. From Seville you can take day trips to Ronda, via bus and an excellent convenient experience, and train or bus to Cordoba. Rent a two bedroom apt in Seville and enjoy the extra room, privacy, kitchen, hopefully a terrace and not having to unpack multiple times. Examine distances using Michelin online maps and you will comprehend how close you are to Ronda and Cordova via a base in Seville.

I like your plan. I don't see much advantage to staying in Seville and day-tripping. Without a car you won't be able to explore the white towns (pueblos blancos) to and/or from Ronda, and with long day trips from Seville, an apartment with a kitchen is not much of an advantage. Also you wouldn't be able to enjoy the evenings in Cordoba and Ronda, with long rides back to Seville. Meals aren't expensive and would you want to cook? If you can get a good connection in Madrid to Granada, then you may save a couple hours of travel time by going straight there, but if you land in Madrid jetlagged and tired after a long-haul flight to Madrid, I think you're better off spending your first days in more laid-back Cordoba than intensive Granada. The train connections are pretty easy, not more hassle than getting through the airport (and hoping your luggage makes the connecting flight) and then from the airport to the hotel. A taxi from the Cordoba train station to your hotel will probably be less than €10.

I'd plan to leave Cordoba in the morning for Granada, go to the Alhambra's night tour of the Nasrid Palaces, then go back to the Alhambra in the morning for a daytime visit, and see a bit more of Granada with any time you have left. That evening you may want to catch a flamenco performance.

On your second evening in Cordoba, see the horse show at the Royal Stables. It's about one hour.

Thanks for both of your replies. We will incorporate advice from both suggestions. At this point, our plan is to fly directly into Granada and stay four nights. We will then rent a car and visit Ronda and Arcos on the way to Seville. We will spend 4 nights in Seville. Depending on how things go, we may take the train to Córdoba for a day from Seville. Also, depending on how things go in Granada, we may rent the car the day before we leave and do a day trip visiting the coast, or some other site (suggestions?). Any other advice would be appreciated.

Actually, I agreed more with Chani's recommendations. I personally am not a fan of Granada (other than the Alhambra, which is amazing). The idea of spending four nights there is not much fun for me, but others will disagree. On the other hand, I love Cordoba and think at least one night there is well worth it. If you arrived later in the day, 2 nights would be great.

Even if flying into Granada, you could still take the train or bus to Cordoba and then on to Sevilla and the White Hills.

FYI - once you do decide, you need to make your reservations for Alhambra in advance. June is extremely popular and so I'd book by April. If taking the train, you can get steep advanced discounts on the RENFE website (not easy to use but not too bad if you use PayPal for the purchase). And to drive legally in Spain, the driver must have an International Drivers Permit, obtained in the US from AAA.

Generally, your itinerary looks sound. But since you are exiting Andalusia west to Southern Portugal, I would recommend beginning your trip in Granada, the easternmost city on your itinerary. The featured visits to the Alhambra in Granada begin at 8:30 AM, so that tour should take place on your second day in Granada. Personally, I would spend 2 nights in Granada, walking through the Albaicin and the old city the afternoon of your morning visit to the Alhambra. It only takes a couple of hours to get from Granada to Cordoba by train or car the afternoon of your second or third day in Granada (drove the Caliphate Road from Granada to Cordoba last May). La Mezquita in Cordoba can be visited in the afternoon. The Que begins at 13:00 or 14:00. La Mezquita was under-rated and a must visit but can't see spending more than 1 night in Cordoba. Cordoba to Seville is less than 2 hours in transit. Would advise a minimum of 2 nights in Seville. And I can't imagine spending all day on a bus from Seville to Ronda and back. Rent a car in Seville, drive the Route of the White Villages, spend the night in Ronda, and drive back to Seville ( 2 hours each way without a stop, but in reality you need to stop in villages such as Grazalema, etc) You could extend your return trip to Seville by passing through Jerez de la Frontera, the famous sherry area of Spain.

Ed

Your general itinerary I think is great. I'm going to jump in here as well and say that 4 nights in Granada is probably too many. I had three nights and two full days in Granada and that was more than I needed. A good day for the Alhambra (buy your tickets online ASAP), and another day to see the city is plenty. Granada is interesting, very distinctly it's own place. It's also very workaday, and aside from the Alhambra and the Albaicin I found the sights kind of underwhelming. I took a walking tour, which helped bring out some of the history as a lot of the old buildings are incorporated in the main town. I dunno, maybe it's modern without being nice, and old without being interesting? Sevilla and Córdoba just have a bit more to offer a tourist. You could love doing absolutely nothing in either town for a day, but I don't think you could say the same for Granada.