Did it last summer, similar age kids. In our case my husband and I each had a carry on size roller bag and a personal item (computer bag for him, camera purse for me). The kids had a similar size roller each - with a small tote bag inside for books and entertainment like ipod so they could pull it out on the plane/train but only had to worry about one bag. It was stressed to them that their bag was their responsibility yadda yadda, and they did pretty well with this.
My kids don't care about clothes so they had a very simple 10-item wardrobe - 4 pants/shorts, 6 shirts that match all the pants/shorts, and a swimsuit. 6 pairs underwear and socks and comfortable shoes. Then the toys/books/electronics that fit in a tote bag and done.
With kids you may want to look into apartments or apartment-hotels, many of which come with a washer. Not often a dryer, so leave time to hang things in the bathroom to dry. If you stay in hotels instead you can find laundry places or services, so this won't be an issue.
It will be much easier to have small bags - trying to lug multiple bags would be a much harder problem - small elevators, small train aisles, longer walks, small car trunks...all of this makes traveling heavy really, really difficult. I did it when my kids were in diapers, strollers, carseats and it was such a pain. This way is much, much easier even though we do need to do laundry.
Barcelona to Madrid is best by train (3 hours) unless you want to spend a few nights exploring on the way, then drive. The drive is much longer - I want to say 7 hours but I don't remember exactly. Train is super-easy.