Cities to experience the best of Spain.

This will give you a taste of Catalunya and Spain. Your limited time would preclude including Andalusia in your plans. Besides, this is best done outside the peak of summer.

Day
1. ARRIVE Barcelona (4N)
2. Barcelona
3. Barcelona
4. Barcelona
5. Late afternoon train (2hr 30min) to Madrid (5N)
6. Madrid
7. Day to Toledo
8. Madrid
9. Day to Segovia
10. DEPART Madrid

When booking flights to Spain, search multi city fares, in to Barcelona, and out of Madrid, or vice versa.

Barcelona has much of interest to me, but I'm not clear exactly what you mean by "culture", since that will be in just about any city or town that isn't given over totally to tourists. I think you'll certainly enjoy exploring the Barri Gotic [edited to add: and La Sagrada Familia], and I really liked the 2-hour tour of that area offered by the tourist office. There are competing commercial tours as well, but the ones I observed seemed to be larger.

I like David's proposed time allocation. Should the bustle in Barcelona get to you before the end of your stay there, you could take a day-trip to Montserrat (which might be cooler) or to the lovely historic city of Girona. Girona has a wall you can walk on, a good museum of very manageable size, two nice churches and a highly walkable historic center.

You will save money on express-train tickets by buying them early, once you are absolutely certain of your itinerary and schedule. For David's itinerary, that would be primarily Barcelona-Madrid. There is some savings possible on Madrid-Segovia, but it's a short trip, so the fare difference is not huge. Madrid-Toledo ticket prices do not change, so you don't need to buy them before leaving for Spain. I think you'll score considerable savings by purchasing round trip tickets to Toledo and Segovia rather than separate one-way tickets. Barcelona-Girona is in the same category as Madrid-Segovia.

What's your tolerance for heat? Madrid (and surrounds) can easily be in triple digits. I'd stick to the east - maybe a couple-three nights in the north (Pyrenees). Girona has a well-preserved medieval center. There's Dali in Figueres and Cadafalque. These can also be day trips from Barcelona. Personally, I would be happy with 9 nights in Barcelona. I never know when people don't want to spend a lot of time in museums if they mean indoor experiences or art. There are some small interesting museums in Barcelona, ancient Roman excavations, folk/ethnography, hash, chocolate. I enjoyed Valencia, can't say that it is much different.

When saying absorb the culture I guess we would mean taking time to slow down and "do as the Romans do" sort of thing. Have a quintessential Spanish afternoon/evening. Definitely do not want to be shuttling from one site to another.
As for tolerance of heat we are both from the south and won't melt in the higher temps. Indoor experiences are just fine. We typically like to have a more unstructured day where we can wander and see what we come across.
Aside from Barcelona/Madrid are there any other areas we should consider? Seville and other areas of Andalusia seem to be well regarded by everyone we have heard from.

With nine days the most I'd want to attempt would be Barcelona, Madrid and a side-trip to Toledo. (Actually, I spent 9 days just in Barcelona last year.) The highlights of Andalucía are Seville, Cordoba and Granada. Please take a serious look at historical weather data before contemplating a trip to that area in the summer. Believe, it's likely to be uncomfortable enough in Madrid without going farther south. Being a tourist is not the same as being at home, where you probably mostly go from your air-conditioned house to your air-conditioned car to some air-conditioned destination. Imagine, instead, spending the better part of the day outdoors, mowing the lawn or gardening. There is a reason why many places in Spain close for a large chunk of the afternoon. Not every indoor location is air conditioned.

Wunderground.com has day-by-day weather stats. How does 16 days with highs of 100+F last July sound? There were only six days that month when the temperature didn't hit 90F. There were only two days with highs below 85F.

If the Madrid + Barcelona combo isn't appealing, you could head to the Basque Country on the northern coast. The weather is usually moderate there.

I agree that Madrid and Barcelona would be a good choice. Summer in Spain will be hot as others have indicated, you may be used to heat but you won't be moving between your air conditioned car and air conditioned interiors. You will be out in the sun for extended periods and even the metro can be hot in big cities in the summer. Madrid is hot but it's at altitude and on the drier side. Barcelona can be humid, when my wife and I were there in September 2005 it was miserable enough that I broke down and bought some shorts. Check timeanddate.com for climate averages. I definitely would avoid the south of Spain in the summer. Also because you would need more time to see that, there is so much there. Madrid has so many great day trips, Toledo, Segovia, El Escorial. With another day or two I'd say spend a couple of nights in Toledo.

I would throw in a different option from other posters. I certainly agree with spending 3-4 nights in Barcelona. It's one of my favorite cities in all of Europe, possibly #2 after Paris. I'm going back in May for 10 days. However, I think Barcelona and Madrid are a bit too similar in size and scope. I'd suggest after Barcelona heading down to Andalucía, especially if you're OK with the heat in the summer. I'd split your remaining days between Granada and Seville. If you did 4 nights in Barcelona, maybe 2 in Granada and 3 in Seville. It's a totally different feel from either Barcelona or Madrid, and it would give you a more rounded experience in Spain. It's also a different culture and cuisine than in the north. I visited both Granada and Seville last summer (even in the heat) and I'm so glad I did!!

If you are ok with very hot weather I would consider the Andalucia option - Seville and Granada. But it is hot, so make sure you get AC!!

Madrid is nice but very heavy on Museums, less good (but still fun) for just wandering and enjoying the atmosphere. Sounds like it maybe shouldn’t be your top priority.

Barcelona is great for what you mention. Walk the barri gotic, the food markets, and go in a few of the top buildings, like Sagrada Familia. Also other neighborhoods. It might be a good city for a food tour. The beach is also pleasant if you like that.

Then hop down to Seville. It will be hot - take siesta and go out early and late. This is quintessential Spain with tapas, Flamenco, moorish architecture, orange trees..
And Granada - my personal favorite (and slightly less hot because it’s in the mountains). But book your Alhambra ticket early. Spend the rest of the time visiting the Albayzin neighborhood which is totally unique and I find it magical.