Leaving Oct. 15. Have 2 days prior and 3 days after following 7-day tour. What to see/do during my free solo days (before, during, after the tour)?
Madrid to Sevilla (with stop at Cordoba)
2 nights in Seville
Torremolinos 2 nights (with one day free from here)
Granada 1 nite
back to Madrid via Toledo.

So many possibilities.
To Segovia prior to tour?
Royal Palace before tour? (Tour incl. Prado)
To Arcos from Torremolinos for the day?
El Rastro on Sunday after tour?
Back to Toledo after tour? or
Flight to/from 2 days in Barcelona after tour?

Suggestions please

I frequent fleamarkets here in the U.S. so I don't bother with the Rastro. My favorite Sunday AM activity in Madrid is to stroll thru (or sit and relax in) Retiro Park: enter from the Puerta de Alcala. If the weather's nice there will be many, many street performers (musicians, mimes, jugglers, etc.). We even sat and listened to a cantaor de flamenco one sunny Sunday morning. The activities start around 11 AM.

Are you allowing enough time for Madrid itself? The Palacio Real is definitely worth a visit (and what it costs for the tour). They offer tours in Spanish and English. After the Palacio Real you could spend some time across the Avenida at the Plaza de Oriente. Then over to Plaza de la Opera, up the Calle Mayor to the Plaza Mayor and continue on to the Puerta del Sol. You will have just hit the high points of Madrid de los Bourbones. If you go down the steps of the Arco de Cuchilleros from the Plaza Mayor you will pass las Cuevas de Luis Candelas (restaurant) and at the foot you will be very close to the world famous Botin's restaurant (los soberinos de Botin). Then you will be in Old Madrid (Madrid de los Austrias). I don't know if Botin's is open for lunch but las Cuevas is. But my preference for an inexpensive lunch is la Argentina, a Tavern just across the Calle Mayor from the Ayuntamiento (City Hall).

The Centro de Arte de la Reina Sofia (near Puerta de Atocha) is a must as is the Thyssen collection.

Daytrips to Toledo and Segovia are definitely recommended (if you have time) and you may want to take a 'Madrid by night' tour. For each of those excursions, I recommend that you take a tour bus. You are more likely to encounter another 'single' on such a tour.

NEDSIRELAND: Thank you for your great suggestions. I will have 2 1/2 days on own and 1/2 on tour in Madrid and was thinking of using one of those days to see Segovia. If that seems silly, I have 3 days after tour in which to do more of Madrid, Segovia, more of Toledo, or quick visit to Barcelona via plane. In Madrid, I plan to see Royal Palace, Reina Sofia, Parque del Retiro and Plaza Mayor. I'm not big on museums, prefering to explore people and places in interesting neighborhoods. I was planning to do Segovia and/or back to Toledo on my own. Are you recommending against traveling to and visiting those cities on my own? Also, if you have been to Torremolinos, what do you recommend I do with one free day in tour? Trek to Arcos?
Thanks again for your help. I'm really looking forward to my first visit to Spain.

Please, when in Madrid, do yourself a big favor and DON'T miss the Prado. I didn't see it on your list, so please make sure your add it. The Sofia is wonderful too, but they're next to each other and both are (IMO) the highlights of Madrid.

Have a great trip!

The tour company agrees, and includes the Prado in the tour itinerary.
Suggestions for other visits in or outside Madrid that give a taste of Spain? What was your most favorite place to which I can easily get from Madrid?
Thank you!