Barcelona is likely to have milder weather than Madrid in March, if that matters to you, but temperature averages suggest that even Madrid is not likely to be very cold. You can get to Madrid easily by AVE train if you wish. Madrid has 3 important art museums, #1 being the Prado, and a large royal palace. I'm in the camp that--aside from the art museums--much prefers Barcelona to Madrid. It's a matter of taste and what interests you. Madrid does have some truly fabulous, and varied, side-trips, of which Toledo is my favorite. I also like Segovia and Cuenca. I'm not sure how much the atmosphere in those smaller cities would be affected by the shorter daylight hours and cooler weather in March, however.
I suggest you get a guidebook and do some reading. See what sounds intriguing to you. If your daughter is free during the period of your visit, I imagine she would prefer to see another part of Spain since she'll have lots of time in Barcelona herself. It will be warmer in Andalucía than in either Madrid or Barcelona. I'd be tempted to head to some combination of Seville, Cordoba and Granada, depending on how much time I wanted to spend outside Barcelona.