What are the classic things to do in Cordoba

The main sites are all in close proximity. In addition to the two you mentioned, there is the Roman bridge and Calahorra tower. There is a bull fighting museum that was nice. The streets in the fairly small historic center are nice to wander, especially if your there while the flowers are in bloom, and doesn’t take very long. The Mezquita is essentially one big room.

Worthwhile sights/attractions:
Alcazar
Mezquita
Triumphal Arch & Plague Monument
Roman Bridge
Museum of Al-Anadalus Life
Calahorra Tower
Jewish Cordoba
Casa de Sefarad
Sinagoga
Artisan Market
city walls
statues of Seneca, Mamonides & Averroes
Bullfighting Museum
Baths of the Caliphate
Plaza de las Tendillas
Roman temple
Museo Julio Romero
Palacio de Viana

The Meziquita is #1. I believe we were able to get audio guides, which I highly recommend. To say that it is one big room doesn’t begin to describe it. I find it really hard to know what I’m seeing without a live or recorded guide. If, like us, you are not well-acquainted with Islamic history and beliefs, some kind of guide will definitely enhance your experience.
As you walk through the streets, peek into the patios of private homes. Córdoba is famous for its private patios, and it is expected that tourists will stop to admire them.

Maybe the Mezquita has improved its audio guide. The first time I was there, about 5-6 years ago, I didn't get one because of the poor reviews. People I met who were using it found it difficult/confusing and not very interesting. You can buy a guide book at one of the souvenir shops which surround the Mezquita - and it makes a nice souvenir too. Or just enjoy its beauty.

The Royal Stables have horse shows in the early evening several nights a week. I was underwhelmed by the Alcazar. I believe the gardens are lovely, but I was there in winter after severe pruning. Other sights I really enjoyed: Casa de Sefarad, Palacio de Viana, the museum in the Calahorra tower (and good view from the rooftop).

Restaurants I go back to for dinner are Taberna La Montillana and Casa Rubio; for lunch Taberna del Rio.

I'll add Museo Archeologico
Visit Mesquita in the first free hour
Spend time on the bridge if it is a nice day and view Cordoba from the other side
Be sure to check the day of the week. Mondays can be more challenging

What time do you arrive in Cordoba? Will you have time for sightseeing when you arrive?

We spent 2 nights in Cordoba, too, and loved it! I wish we had a 3rd night. According to my notes, this is what we did:

We arrived in Cordoba from Toledo in the early afternoon. After checking into our hotel and having a quick lunch, we decided to visit the Mezquita, arriving there at 3:30PM. There were no lines. We rented the audioguides. At first it was a little confusing because I didn't know where I was in relation to the audioguide. If I remember correctly, I don't think there is any signage inside the Mezquita so you have to figure out where you are based on the audioguide text. I think the audioguide is worth it. Perhaps the guidebook that Chani mentions is worth it, too. You will appreciate and understand the Mezquita better if you have some sort of guide, either the audioguide, guidebook, or live guide.

That night we had dinner at Regadera, Ronda de Isada 10, on the river bank. It was one of the best meals of our trip. You have to make reservations in advance.

After dinner, we walked across the Roman Bridge, all lit up, and turned to see the Mezuita lit up as well. A stunning view!

The next day we visited the Alcazar, which I highly recommend. This was in September, and the gardens are in full bloom and gorgeous! There are pools, fountains, lush greenery and beautiful flowers. Make sure you go to the top of the tower for lovely views of the gardens.

After the Alcazar, we took a taxi to Palacio de Viana, a small palace with 12 wonderful patios. This is a hidden gem. Then back to the Juderia to explore, but I wish we had more time for the Juderia.

Our dinner was at Mercado Victoria in the newer part of Cordoba. I highly recommend!

Hope this helps.

For one of your evenings, I'd take this walk that covers a lot of outside sights, tapas places and, when you ready for dinner, restaurants. Thus if you head from the Mezquita-Cathedral, to Plaza de la Corredera, then towards the river via Plaza del Porto, then along the river bank back towards the puente Romano.

The Mesquita was the top sight for me. The old quarter around that area had the rest of the sights we were interested in. I liked the small Jewish synagogue but skipped the Alcazar.

During our visit to beautiful Córdoba we enjoyed watching the Royal Equestrian Show, “Pasión y Duende del Caballo Andaluz” ("Passion and Spirit of the Andalusian horse") at the Royal Stables. You can find more information here:

• Cordobaecuestre.com
• caballerizasreales.com

We also enjoyed visiting the Patios in Barrio San Basilio. We bought our tickets at their tiny office, where you get a map with the route of the patios. The home owners are very proud of their patios, and they are happy to share stories about how the patios came to be.

We speak Spanish, but most of the home owners don't speak English (except for the young ones).
When my mom and I were there, we noticed a couple of women who were taking a tour with an English speaking tour guide, so perhaps you can contact them and ask about this possibility.

Enjoy your visit to one of my favorite cities in Spain!

Edited to ad...
You'll find more information in my trip report: Our Trip to Spain.

I can suggest two other options: Just outside the city is the largest archaeological site in Spain, the Medina Azahara. This was a tenth century fortified Arab palace/city. There is a museum as well as the extensive ruins to explore. A bus runs directly between Cordoba and the site. My other suggestion is a change of pace after travel and tourism: Try out the Arab baths at Banos Arabes y Hospederia. They are refreshing and relaxing.