If you wanted a smaller but not small city to stop in between Barcelona and Madrid check out Zaragoza - a real gem right on the main rail line:

https://www.google.com/search?q=zara...HfLrB0kQsAQIHA

Stay say 2 nights - 4 nights in Madrid with one day trip to Toledo - Segovia is nice but Zaragoza IME even much nicer.

Barcelona is a good idea for (possibly )some beach time and to experience a completely
different city and culture than Madrid.
Easy tran ride to Madrid 2:30-3:00 hours.
From Madrid , you can visit smaller towns -Segovia and Toledo- only 30 min by train ( day trips.?)
With 2 kids, you might look into renting an apt. in Madrid , we rented from Spain Select three times. In Barcelona , we rented from Habitat several times.
I would leave Andalusia for another time.

Madrid and Seville would be my choice. High speed train between Madrid and Seville. However, it will be warm then, so perhaps Barcelona would be more comfortable.

Yup weather starts to boil in Seville in June - avg hi 90F:

http://www.holiday-weather.com/seville/averages/june/