Advice for 1 week in Spain.

I will be in Spain alone in May and then will my spouse in September. I will tell you more when I return.

What I do know is that hotels in Barcelona and Madrid are not cheap, esp. the former. Also, 7 days is not a whole lot and if you need some time to recover from jet lag, then I suggest you lengthen your stay.

Spain is overflowing with interesting areas and 7 days doesn't get you much. You don't state what interests you. Some would just stick around Barcelona and do day trips from there. Madrid will be considerably cooler than the southern coast. It is a common addition to Barcelona if you must try a second city. But as mentioned, they are more costly.
If you can arrive in Seville or Malaga, then you have the options to visit 2 or 3 cities in Andalucia. It is bound to require an aditional 'leg' to your flights and may cost more, but it gets you where you want to be. There is a considerable distance from say Seville to Barcelona unless you fly. Then you need to add in all the time to and from airport, wait times as well as flight.
Andaluscia will be cheaper in November than October, low tourism, numerous 'hostals' (these are 2 star family run hotels; not hostels) and most likely the best bang for your buck and time. You could do Seville, Granada and Cordoba. Not as full a visit to all three, but certainly the highlights.

Good replies so far. You decisions depend a lot on personal interests. Many Rick Steves fans love smaller towns, but getting around by bus can mean a lot of waiting around. I think Barcelona (a wonderful destination) is over-sold on this forum. It's isolated enough that you might prefer Madrid and Seville, or just one of them, and the surrounding attractions. Prompt, affordable flights from Denver (?) are a big factor in this decision. Another part of my point is that when you only have seven days (which means 5 1/2 days, more or less) every change of bed means huge percentages of your trip lost to moving. I personally don't care as much for the far south, but I think if you played golf, you'd want to go there. (?) We have been to Madrid several times, and have not exhausted the possibilities yet.

There are parts of Spain that are so attractive, but just not practical for low-budget 5 1/2 day trips, like the White Villages. I have not been to Rick's favorites on the northeast coast.

Ummmm....Norwegian Air Shuttle hasn't released their November schedule, yet, but they do fly Denver to Barcelona with one stop (London Gatwick 6 1/2 hour layover). Their latest posted flight (at the moment) is late September (26th) at a cost of $241.70. This is their economy fare - 10kg carry-on luggage, no meals served, no seat selection. I'm guessing November flights will be lower. I checked Trivago - lots of :highly recommended hotels in Barca for around $100/night - I randomly plugged in mid week arrival and a three day stay.
You could AVE (fast rail) from Barca to Madrid, and see Spain's two big cities. I didn't check on hotels, there, but I suspect you could find something in a similar price range.
No limited amount of time in Spain is satisfactory....a week, a month - it's a wonderful world over there, you'll see.

As you know, or will soon discover, the flight options from DIA to Europen destinations have been 1) Lufthansa, connecting thru Germany, 2) British Airways, connecting thru London Heathrow airport, 3) United, often with a connection somewhere in the USA, like Chicago, 4) Icelandair, connecting thru Reykjavik, and 5) Norwegian Air, connecting thru London Gatwick airport. The last two, relative newcomers and offering super cheap fares, especially if you carry on small and lightweight luggage and aren’t looking for frills, seem to be causing the first three to change some of the ways they’ve operated over the past 20 years.

Our first trip to Spain stuck to the southern part, (Andalusia) with a side-trip into Portugal, taking 3 weeks. Our second trip, for 2 weeks in November 2012, included Madrid, Segovia, Toledo, then north to Bilbao, San Sebastián, Girona, and Barcelona. We needed the whole 2 weeks for that trip, though. We landed on the day of Madrid’s exclusive Fiesta de la Almudena, a religious event, and caught a Zarzuela performance the next night. Zarzuela’s a Madrid-only musical spectacle, and November is in Zarzuela season. It wasn’t freezing cold in November, but a jacket was needed throughout that Trip. Nights and mornings were cool, and we got wind and rain some days and nights.

If you based yourself in Madrid, in addition to outstanding art museums (the Prado, of course, and the Reina Sophia, but also the surprisingly wonderful Thiessen-Bornemisza) within the city, and Rick’s city walk described in his guidebook, Toledo and Segovia made for great day trips. Squeezing Barcelona into your trip would be possible, but with only 1 week, you’d get more bang for your time and buck sticking with Madrid and a day trip or two from there.

Or, if warmish temperatures are important, confine yourself to Andalusia, and visit Granada (Alhambra Palace) and Seville, maybe also Ronda and/or Arcos de la Frontera. Won’t guarantee how warm it’ll be, but probably warmer than Evergreen or Idaho Springs at that time of year! We’ve used a combination of train, rental car, and bus on both trips. Driving in the country and smaller towns is doable, but don’t trouble with a car in Madrid, Barcelona, or Seville.

Once you decide where you’re flying into and out of, start researching places to visit in those areas. If you want out of the way places to visit, a rental car is the way to go. There are literally hundreds of churches, hermitages, bridges, etc., very much worth seeing, but there are no bus stops or train stations within miles. It really depends on what you like to do and see.

I'd agree with jaimeelsabio that the priority is to decide what flight options are practical/affordable. If the only direct flights are Barcelona and Madrid then they may be the best bet given your short available time.

But if you have to change anyway in somewhere like London or Amsterdam then you can fly to Malaga or other southern airport just as easily and skip the Spanish north/midlands.

Assuming the flights work, a good 7 day taster introduction to proper Spain might be four nights in Sevilla, with day-trip to Cordoba, followed by a day driving across to Granada (north via Ubeda or south via Ronda) and two nights in Granada. Alternatively three nights in Sevilla, one in Cordoba and then spend a day driving to Granada.

My first visit was to Madrid and Andalucia. Depending on what you want to see, I'd go with a few days in Madrid, then on to Cordoba and Granada.

While Madrid may be colder, the airfares might be cheaper and the hotel prices are lower on average than Barcelona.

Heather and others are right about the weather differences between parts of Spain.

However, in temperature terms, I'm not sure I'd make it a deciding factor for November (as opposed to August when extreme heat can make some parts best avoided if you have a choice). Admitedly, I know nothing about Colorado beyond watching Mork & Mindy. But according to Google, November night lows are already dropping below zero and daytime highs not much above 10°C in parts of your state. Compared to that you'll find anywhere in mainland Spain "warmish". In Andalucia and Murcia highs will still be over 20°C in the day for much of the month, but even in Madrid it will be nearer 15°C. Freezing overnight would be unusual even in the central parts and very unusual in the coastal provinces.

What you might consider, though, is rain. Don't be surprised if it rains a bit most days along the northern (Atlantic) seaside for example.

For a single traveler, trains are less expensive than cars. Cheap is a relative term. November is low season, so prices for rooms are relatively low. Spain is generally less expensive than Rome/London/Paris and Israel. Since you are 60+ you can get a tarjeta dorada "TJ" once you arrive in Spain for €6. With that, you can get 25% or 40% discounts on full-fare train tickets on AVE and AVANT trains. Or you can buy train tickets with even bigger discounts (but no refund/no exchange) when they go on sale, usually 62 days in advance. Since you are 65+ you can get discounts at a lot of sights.

With only a week, I recommend either Barcelona or Madrid/Andalucia. Barcelona has enough to fill a week, including day trips to smaller towns (Girona, Figueres). It doesn't look like there are non-stop flights between Denver and Spain, so if you choose Andalucia, consider Sevilla, Granada and/or Malaga. A week in Andalucia isn't much time as there's a lot to see and do. If you are a fan of European painting, you'll want a couple of days in Madrid for the art museums, otherwise skip it - Andalucia is more interesting with smaller towns, centuries more history and charm.

Just agreeing with the other replies. With only a week, pick one region, or at most two regions.

As for flights, since you have to change anyway, you can go anywhere (although of course it may cost you much more to go to some places than others). Note that if you are interested in the south of Spain, Malaga is a much larger airport than Seville, so you may have more options and/or lower prices if you fly there.

As for Spanish hotel prices, I agree with Chani and disagree wtih Barkinpark - I find them reasonable (Barcelona) to downright cheap (Madrid and Andalucia). If you don't need a fancy place, investigate hostales. An hostal is not a hostel - it's what in other countries would be called a guest house or pension.

For instance, I stayed in the Hostal Acapulco in the very center of Madrid in 2015 for €57 per night! http://www.hostalacapulco.com/ (if it shows up in Spanish, click the British flag for English). Note that while I made my reservation here in English by email with no difficulty, the desk staff often spoke only Spanish. If you need English speaking desk staff, you may want to stay elsewhere. My room was small but fine for one person.

In Toledo, I stayed at the Hotel Santa Isabel for €30 (€35 since I had breakfast): https://www.hotelsantaisabeltoledo.es/. Here, while the website is only in Spanish, the staff spoke some English (again, I made my reservation by email in English with no problems). The room was nice, but the roof deck was even nicer.

In Barcelona, my sister stayed recently at the BCN Fashion House: http://www.bcnfashionhouse.com/. She had a bathroom down the hall, but paid something like €65 per night. Her only issue was that the desk was reliably staffed only in the morning during breakfast; if she had questions later in the day, she was out of luck. English was spoken (I know for sure, as she speaks no Spanish or Catalan!). She otherwise had a good stay, and the location is great - very close to the center, but a bit out of the way so it's not "tourist central."

I just give these as examples to show what you can get on a low budget - not perfection, but perfectly acceptable, for a fraction of what you'd spend in most other cities of similar size for similar accommodations. For more ideas, look at EuroCheapo: https://www.eurocheapo.com/

Also, even on a budget, taxis in Spain are inexpensive and (in my experience) very reliable. They use the meter, don't take the "scenic route" to drive up prices, and any extra charges are legitimate and are posted in the taxi (for instance, for putting things in the trunk or picking up at a train station). So, even as a solo traveler on a budget, they're worth considering to get between your hotel and the train station, bus station, or airport - particularly if tired or jet lagged, or in a place like Toledo where the walk from the stations is harder than it looks at first.