Itinerary late May

Madrid is much more than just 3 art museums lol! In fact it is one of the Art Deco capitals of Spain along with Valencia, plus it has the best Archaeological Museum in Spain too. What's problematic is you have 2 nights only in Barcelona, if possible I'd actually drop Barcelona and add a night to Cordoba and Sevilla. If you are already stuck with this open jaw plan - I'd drop Granada and Cordoba, add two nights to Barcelona and one to Sevilla (with day trip to Cordoba), while not ideal, I feel it's an improvement from your initial literary, where every stop is lack at least 1 night, except perhaps Madrid and Granada.

So something along these lines:

Barcelona (4 nights)
Madrid (2 nights) - or switch out for Valencia for Art Deco
Sevilla (4 nights) - day trip to Cordoba

So no Alhambra?

If you would like to include Granada, I think you can try flying direct from Barcelona to Granada, drop Madrid, then continue from there, so like this:

Barcelona (4 nights)
fly to
Granada (2 nights)
AVE train to
Cordoba (1 night)
AVE train to
Sevilla (3 nights)

Lol! But now that I've looked up Valencia, I think I want to go there!! From Barcelona, should I fly, drive or train?

There is a direct train from Barcelona to Valencia, I'd just use that.

Whilst it's simple to get from Barcelona to Valencia, it's not so easy to go onto Granada.

Given your limited time and entry (Barcelona), and exit (Sevilla), points, I'd skip Madrid this time for the reasons you've given (and also skip Valencia, though admittedly it is delightful).

But my preferred holiday is seeing fewer places for longer, so this might not apply to you. For what its worth Barcelona - fly Granada - railway via Cordoba to Sevilla. 4-2-4 nights.

P.S. There is a fair amount of art deco architecture in Sevilla. The city is more famous for its moorish, mudejar and gothic sights, but the modernist exists too, for example in the 20's Expo buildings.

You have nowhere near enough time to see all the places you've mentioned. If you try to spread yourselves that thin, all you'll accomplish is spending a great deal of time on trains and planes and getting between your hotels and the train stations/airports. The prospect of having only one full, non-jetlagged day in Barcelona is frightening.

You're going to need to go back to Spain. Put together an itinerary for this year that allows you adequate time for a shorter list of destinations. There are other very interesting destinations you should consider for trip #2, primary among them Toledo.

The Valencia-Granada link is to be avoided on a short (or medium-length) trip unless you want to waste time dealing with airports. (I haven't verified that there are flights between the two cities.)

O.K. Madrid is out.

Barcelona 3 nights
Cordoba 2 nights
Granada 2 nights
Seville 3 nights

And I may still change that....

That's better but you're still short of time in Barcelona, where your first (partial) day may be severely compromised by sleep-deprivation and jetlag. There are six highly popular sights that require pre-purchased tickets to avoid insanely long ticket lines and possible sell-outs. Those tickets are normally timed as well as dated, so it's difficult to cram a lot of places into each day. How will you know how much time to leave between Sight 1 and Sight 2? That requires estimating how long you'll wait in line (despite having a ticket), how long you'll want to spend inside, how long it will take you to get to the next sight and (at some point during the day) how long you'll spend at lunch. It's like putting together a jigsaw puzzle.

If you are that unusual visitor with little to no interest in the Gaudi and other modernista sights, you may be OK with 3 nights in Barcelona.

I think I will enjoy the Gaudi. Is there any way to skip the lines? I haven't purchased any attraction tickets yet.

You need to buy tickets in advance for any of these that you want to see: La Sagrada Familia, Parc Guell (no tickets sold at the park), Casa Mila/La Pedrera, Casa Batllo, Picasso Museum, and possibly the Palau de la Musica Catalana (English tours can sell out). In general, everyone with a ticket for a specific time frame lines up, then people are allowed to enter in groups as earlier visitors leave (or so it appears--I'm guessing the fire marshall insists). Obviously, the first time slot is a choice option, because there's no risk you'll be tied up elsewhere and unable to get there on time, and there will be no waiting for earlier visitors to leave; you may still have to wait a bit to avoid crowding at a bottleneck (thinking of the two Casas here).

At some sights you can pay extra for a sort of wildcard ticket that supposedly allows you just to show up (perhaps on a specific date, I'm not sure) and enter without waiting. I assume that means you go right to the head of the line, but I haven't bought one of those tickets. The plain-vanilla tickets are already quite expensive.

The good news is that many of the popular sights have hours that extend into the evening, which allows you more than 8 sightseeing hours per day if you don't run out of energy.

The Sant Pau modernista site is really lovely (same architect as Palau de la Musica Catalana) and is that rare bird in Barcelona: a place where you can (up till now) walk right up and buy a ticket for immediate entry. There are also many museums in Barcelona that generally do not have entry lines of any significance. But it seems that a large percentage of tourists want to see many if not all the places I listed above. I'd also want to allow time to walk around the Eixample (to see some of the modernista buildings that are not open to the public) and the medieval Barri Gotic.

Thanks for the ticket information. I will look at that now.

I'm not sure where the line is (or if there is one) between Modernism and Art Deco. Barcelona's modernisme can fill several days.

Thanks for all of the advice. The trip is now:

Barcelona 4
Granada 2
Seville 4

And I suspect I will return to Spain.

With the time you have available, I could not improve on that time allocation. Going back is highly recommended.

You suspect, I'd lay money on it :-)

Good job with the itinerary. Day trip to Cordoba is a must, for the unique and stunning Mezquita and the excellent Juderia.