15 days Itinerary in Spain

The itinerary looks great. Personally, I'd take a night from Barcelona and add it to Granada or Sevilla (to include day-trip to Cordoba), but that's just my preference and you certainly won't get bored in Barcelona either.

Vueling is fine. Make sure you understand the rules on, for example, any luggage you are bringing (Basic for example means, I believe, no hold baggage included in the price).

Renfe website can be hit and miss in getting payment to go through with some cards. If you have a problem, try Trainline instead. Depending on when you're visiting and how remedial work progresses you may find the train on day nine involves a coach for the first part (to Antequera), then rail for the remainder. An alternative is to take an Alsa coach for the whole journey which should be quicker.

PS I don't know if it counts as budget but in Granada we like the Posada del Toro hotel. Excellent location, interesting old building with good rooms, adequate breakfast (for Spain), and now provides kettles.

That's very helpful information. Thanks Nick!

Not certain what budget friendly means to you. On out last couple trips to Madrid we stayed at Hotel Europa at the Puerta del Sol. In Sevilla it was Hotel Amadeus. On our last trip to Granada we stayed at the no frills Hotel America right at the Alhambra complex. Admittedly, it’s not for everyone, but we enjoyed it for the 3 nights we stayed there. Getting into town was easy, but may be too far removed from the downtown for many people. Since that was our 4th time visiting Granada, it worked for us.

Your trip looks great for a first taste of Spain, and it seems you've done your research to avoid putting in too much in too short a time.

For budget Barcelona lodging, my sister recently had a good stay in the Fashion House Bed and Breakfast: http://www.bcnfashionhouse.com/. She stayed in a room with bathroom down the hall, but there are some rooms with private bath. The location is great, and she made great use of the supermarket across the street. She was able to walk to the Place Catalunya (handy for the airport bus), Ramblas, Barri Gotic, etc, while being in a somewhat quieter area. It's not in the Eixample, which would be pricier, but again, she was able to walk there easily.

For Madrid lodging, I had a good stay at the Hostal Acapulco: http://www.hostalacapulco.com/ (if the website comes up in Spanish, click the British flag to get it in English). Note that in Spain, an hostal is not a hostel, but rather what would be called a pension or guesthouse in other places. The rooms are small, but perfectly serviceable. The location is very central, but on a quiet side street. Note that while my e-mails to make the reservations were handled in fluent English, many of the staff spoke minimal or no English (I handled most things in Spanish). So, if you want a fluently English-speaking staff to help you during your stay (which I certainly understand), you may prefer to stay elsewhere.

I'm fond of the EuroCheapo website for budget friendly accomodation recommendations; note that not all of their listings are "cheap." Here are their pages for Barcelona https://www.eurocheapo.com/barcelona/, Madrid https://www.eurocheapo.com/madrid/, and Seville https://www.eurocheapo.com/seville/.

Does "early morning" suit your family? How early is early? I try to use evenings for moving. That way I get a full day of sightseeing, then relax on the ride with a picnic supper - or have dinner after I check in to my hotel, especially in Spain where a 9 pm dinner is early.

I flew with Vueling once and had no problems at all. I thought the staff was very friendly. We used trains a fair amount on our recent trip to Spain. I would say that RENFRE is fine. I had problems trying to book tickets in the station (language barrier) and ended up buying any last minute tickets online. I used trainline.eu or loco2. I would suggest purchasing the tickets in advance for price savings. Within Madrid, there was staff at a few of the stations to help with train issues and I typically saw RENFRE staff available for help with directions and obtaining tickets from the machines for the local (Cercanias) trains. I'm not sure how often it happens, but we had a RENFRE train just cancelled on us and the next train was full so we had an unanticipated wait. Also if there are plans to daytrip to Toledo, it seemed like those trains might sometimes fill at the beginning and end of the day.
Cordoba is close to Sevilla and was one of my favorites of the cities we visited. Madrid is very lively, but Cordoba is so much more beautiful, interesting and historic. The Mezquita in Cordoba is an amazing place to visit. I would take a night or two from Madrid and spend some time in Cordoba. Otherwise, I think your itinerary is good.
For hotels, in Barcelona you could try BCN fashion house, in Granada we very much liked Hotel Anacapri. We booked directly with the hotel mentioning RS guide and we got breakfast included in the rate. In Sevilla we stayed at El Rey Moro. It is just beautiful and reasonable with a very nice breakfast. In Madrid, we also highly recommend Hotel Europa. If you do decide to give Cordoba some time, we stayed at the Hotel Mezquita. All these properties were very reasonable with a lot of local charm. (Except for Hotel Europa which though very nice, is a "hotel")

I would also spend just 4 nights in BCN to allow for the Mezquita from Sevilla. If Madrid’s superb art museums are of particular interest then I would probably spend 4 nights there and just 3 in Sevilla.

Day
1. ARRIVE Barcelona (4N)
2. Barcelona
3. Barcelona
4. Barcelona
5. Fly to Granada (2N)
6. Visit the Alhambra
7. Train to Seville (4N)
8. Seville
9. Day to Cordoba for the Mezquita
10. Seville
11. Train to Madrid (3N)
12. Madrid
13. Day to Toledo
14. DEPART Madrid