First time in Spain... Recommendations

It's probably too late to get a spot at Tickets in Barcelona, but it's worth a shot a lunchtime. If not Tickets, then try Bodega 1900.

Also, stop in at Formatgeria La Seu for some cheese & vermut. It's in the Barri Gotic.

Do you have Rick Steves Spain? I found it invaluable for my trip to Madrid and Barcelona last year.

In Barcelona, if you want to see the Picasso Museum, the Sagrada Familia, Casa Batllo, Casa Mila, Park Guell, or The Palace of Catalan Music, you will want to make reservations. In such a short trip, don't even try to see them all, but do pick one or two that interest you and make reservations, to avoid disappointment. I saw the Casa Batllo and the Palace of Catalan Music, and was very happy with my selections.

Note that Puerta del Sol takes no time to see (it's just a busy plaza), but Rick's suggested walk around the center is good (it connects the Puerta del Sol with the Royal Palace and Cathedral, via some interesting small streets).

Barcelona has a large number of sights. You need to get hold of a guidebook and do some reading. Some of the places you decide to visit may need timed tickets to avoid lines. Best to know that ahead of time.

In Madrid, you will want to see the Plaza Mayor, which is right around the corner from Sol and down a block or two. Mercado de San Miguel (outside the Plaza Mayor) is a great place for a break. Google "Madrid eats" and you will get a list of many great Madrid restaurants. Rooftop restaurants are plentiful.

First, Congrats! You have picked a wonderful country and time of year to visit.
Second, the number of days selected for each city is great.
Third, you need to quickly peel back the layers of doing your own research to prioritize where you desire to spend time. Allow me to help you by increasing your time each day by..........
Fourth, on your first day commit to adapting to the culture of time utilized within Spain. This means NOT allowing your American trained feeding habits to dictate when you eat in Spain. Eat tapas early and then a full meal later. Go out for late evening walks (join the families you will find already out walking). Stay out late and get out of the traditional tourist areas to gain anotherperspective of how locals live.
Enjoy!

Day 5 - I highly recommend a day trip to Cordoba to see the Mezquita and walk around the Juderia - the best-preserved medieval neighborhood - which you won't see in your other cities.

I would definitely add a day trip to Córdoba on Day 5.

Our favorite restaurant in Seville was La Paloma Blanca (in the Triana neighborhood). It's not what I would call fancy, but they have a nice wine selection and the food (especially the seafood) is amazing and beautifully presented. Rick's book says they open at 8, but the didn't actually open the doors until 8:30 the nights we ate there (it was so good we ate there twice!).

We made reservations at Bodega 1900 in Barcelona and were blown away. Make sure to try the house vermouth as an aperitif.

If you are planning to visit either the Sagrada Familia or the Picasso museum, I would highly recommend buying advance tickets. The Barcelonetta beach scene was very enjoyable - we spent a few hours there one afternoon and actually felt like we were on vacation!

I agree with others - Spain is a great country - I have only touched the surface in my last two trips there but I love it. An earlier poster mentioned adjusting your eating habits to the Spanish clock - this will really enhance your experience. In Seville - head to the Triana district across the river - not sure about RS's recommendations but I would encourage you to find a spot that looks welcoming and fun. We found a plaza near a church called St. Anna (I believe) that had several restaurants - we picked one and had a great long dinner (3 hours) - I really enjoyed watching Seville families with young kids still out at 11:00 or later. And the walk back to our apartment was romantic and lovely. You don't mention the cathedral of Seville - if you are there for two days do try and go there. It is amazing. And climb the tower!

Enjoy Spain - eat lots of jamon (especially in Barcelona) - its a splurge but well worth it.