You can get to Mallorca from Barcelona for about €40 one way and the flight should average about 45 minutes. Mallorca is a fantastic island, we're often there around twice a year, usually May/June and again in October. It is so varied, my favourite part being the Tramuntana Mountains. There is a fantastic old wooden train that leaves Palma, up through the mountains and into the wonderful port of Soller, one of the best train journeys in Europe: http://trendesoller.com/en/
Palma is a very nice capital city, quite compact and easy to navigate with plenty to see. There are lots of ports and villages dotted around the island so a car is very beneficial. You could easily occupy a whole week on the island and I would spend less time in Barcelona and more in Mallorca. The pace of life is less than the mainland and this makes driving for example a bit easier.
One tip....avoid Magaluf like the plague unless you like the sight of drunken young Brits making absolute fools of themselves.