First Spain trip (11 days) - Madrid, Barcelona, Seville and/or Granada

This is a well worn path over your time frame.

Arrive in Madrid (3 nights)
...with day to Toledo
Train to Sevilla (3 nights)
...with day to Córdoba
Train to Granada (2 nights)
Fly to Barcelona (3 nights)
Depart from Barcelona

Oceantravel,

Keep in mind that you will arrive the day after you depart from the US, so count the number of nights you will be on the ground, not counting travel days.

• It looks like you only have 10 days (11 nights) to work with,
  correct?

• I would suggest visiting 2 major cities.

• You may consider booking, "multi-city" tickets to avoid backtracking.

• You could fly into Madrid & out of Barcelona or vice versa.

• If you're set on visiting Madrid & Barcelona, I'd suggest skipping
  Sevilla & Granada on this trip

• If you want to visit Madrid & Sevilla, then I'd skip Barcelona.

• If you want to visit Sevilla & Granada, then I'd skip Madrid & Barcelona.

• I think your trip will be too rushed if you plan to visit all the
  cities on your list.

Once you narrow down, which cities you want to visit, then you'll decide which day trips are feasible from those cities.

• If you decide you want to visit Madrid, you can take day trips to
  Toledo, &/or Segovia.

• If you decide to visit Barcelona, you can day trip to Girona, &/or
  Montserrat, &/or Figueres.

• If you decide to visit Sevilla, you may take a day trip to Córdoba,
  or spend an overnight on the way to Sevilla or Madrid.

Having said all of this, I'd recommend spending at least one night in Toledo & Córdoba to experience these beautiful cities at night, without the hordes of tourists.

If you decide to travel to Granada, make sure you book your Alhambra tickets in advance ;-)

Keep this in mind...

2 nights in any location is equivalent to one full day in said location.
3 nights = 2 full days
4 nights = 3 full days.

Each time you relocate from one destination to the next, you will lose at least half of that (travel) day.

The day you arrive (from the US), you may be exhausted from jet lag & thus you will not be very productive.

If I had 11 nights, (10 days) to work with, I'd choose something like this:

• Madrid & Barcelona, 5-6 nights each ( 4-5 full days), with possible day trips (as mentioned previously)
• Madrid & Sevilla, 5-6 nights each ( 4-5 full days ) with possible day trips (as mentioned previously)

Enjoy!

" I also don't want to feel like I'm running around too much and can't relax on the trip."

If that's really true, please follow Priscilla's advice, and pick two of the three (Madrid and around, Barcelona and around, Andalucia). If you want a "tasting platter" and don't mind spending a lot of your trip in transit rather than in places, you can use David's itinerary.

Once you've decided which places you are seeing, you can then book flights. If you are starting in Andalucia, it's easy to fly to Seville (will require connections coming from the US, but lots of options). Madrid and Barcelona have nonstops back to the US, which means you can usually leave a bit later (a flight from, say, Granada to Madrid to the US will probably involve a very early departure from Granada). Do book a multi-city flight instead of a roundtrip, to avoid backtracking.

As Harold says you will need to consider what sort of experience you want when deciding how many places you wish to visit.
As per my itinerary above, only 3 out of 10 days spent on the ground will be 'travel days'.
Madrid to Sevilla is 2hr 30min by train.
Sevilla to Granada is 3hr 20min by train.
Granada to Barcelona is a 1hr 20min flight.
Of course flying involves extra time with airport checkin and transit times. Also, don't forget that day tripping involves travelling time, to and from, and that should be taken into account as well.

As you've probably inferred from Priscilla, Harold and David (all good advice), it depends on your travel style. Some people can spend a couple hours on a Hop On, Hop Off bus and another couple hours at one or two sights and feel they've "done" the city. Others want to take time to "wallow" in museums (I'm a wallower), and wander the back alleyways.

How efficient are you at moving around? As David says, train travel is fast in Spain. You'll have to pack up your luggage, check out and get to the station, then go to your next hotel to drop your luggage before you begin sightseeing. You do need to be at the train station at least 15 minutes before departure to find the train platform. For high-speed trains (and those are the ones you'll probably take except possible to Granada or Toledo), you will need a few minutes to go through security where they scan all your belongings. Madrid's train station is very big, Barcelona's isn't small.

With 10 nights, I'd highly recommend no more than 3 overnight stays. Decide which 2 of Madrid, Sevilla and Granada you want to see. Frankly, if you aren't art lovers, I'd suggest skipping Madrid and going straight to Seville after landing.

Thanks to everyone who posted for their advice!

Chani, you bring up an interesting idea. I know I had originally mentioned that Barcelona and Madrid were must-haves in our trip, but if we wanted to focus more on Barcelona and Sevilla and/or Granada and spend less time in Madrid, sounds like another possibility to consider. (We could always come back to Spain to explore Madrid more at a later date)

Expanding on your thoughts above, removing the focus from Madrid, but leaving it as your departure point.

Arrive in Barcelona (4 nights)
Fly to Granada (2 nights)
Train to Sevilla (4 nights)
... with day trip to Córdoba
Early morning train to Madrid (1 night)
... with afternoon for the Royal Palace or Prado
Depart from Madrid

Sevilla to Madrid is 2.5 hours by AVE train. An 08:30am departure would see you in Madrid at 11:00am, giving you an afternoon to enjoy before flying out the next day.

It would be better to book the flats from owners rather than to buy hotels

My thinking is more in line with Priscilla's but I will offer another possibility if Madrid and Barcelona are priorities. Fly into Madrid, spend about 4 days, including a day trip or two (Toledo and Segovia are the most common but there are many other choices). Then Valencia, which is a completely different region and feel and will be the most stereotypical "Spanish" city. It's a smaller city so just one night there is adequate, though you can easily spend two. Easy and quick trip from Madrid by high speed AVE. Train to Barcelona and spend 4-5 days there. While I could spend that entire time in the city, there are some nice daytrip options, including Girona, Figueres (Dali Museum), Montserrat, Sitges (beach town) and Tarragona (Roman ruins). Fly home from Barcelona.

If you want to see Sevilla and Granada, that to me is it's own trip. You could included a couple days in Madrid and still have plenty of time, but Barcelona is not really convenient to Andalucia and should be left for another trip.

Thanks for all the suggestions! We ended up booking a flight into Madrid and will be departing out of Barcelona. We had to leave one day later than originally planned, so we will have a total of 10 days on the ground in Spain (excluding 2 travel days of departing from the US to Spain and departing from Spain back to the US). We will most likely divide our time among Madrid, Sevilla, and Barcelona on this trip, and save Granada for another future trip, due to time limitations. We are very excited! Our general interests include architecture (will definitely plan on seeing some Gaudi), food, some art (Prado and Joan Miro museums are in consideration), history, music, and nightlife. We may also try to explore flamenco shows, and if available, Spanish guitar shows or other concerts. I'm also open to suggestions in Madrid, Sevilla, and Barcelona, that line up with our interests. We bought a couple travel guides and are in the process of figuring out where to go.

Sevilla is a great place to see a Flamenco show.
We enjoyed the one hour show at, 'Casa de Flamenco'. Make sure to buy your tickets at least a day in advance since it sells our rather quickly.

When in Sevilla, make sure you visit, La Plaza de España, as well as the highlights, including the Royal Alcázar & the cathedral. There are so many beautiful monuments & churches there too!

If you're interested in buying ceramics, make sure to go to Triana when in Sevilla.

My favorite place for tapas in Sevilla is 'La Antigua Bodeguita', on Plaza San Salvador right across from the church.

Enjoy!

EDIT: I know you don't have much time, but you may want to stop in Córdoba on your way to Sevilla. You can store your luggage at the bus station (across the street from the train station)
You can take a taxi to the Mezquita, then enjoy a delicious lunch at Bodegas Mezquita or at Casa Pepe de la Judería, before strolling through the Jewish quarter & buying some souvenirs at the small artisan shops, (Zoco Municipal) along Calle Judíos ;-)

I believe there are occasional guitar concerts at the modernista Palau de la Musica Catalana in Barcelona. You could kill two birds with one stone.