Iḿ not sure why you are heading to Barcelona first...Its a way the other side of the Iberian Peninsula!!!!
There is no Airport at Braga. The nearest airport to Braga is Porto which is 55km away.
There are direct flights to Porto from a variety of destinations.
There is a list of current destinations served here:
http://bit.ly/porto_destinations_2010
Although low-cost Ryanair operates Barcelona Girona (Change of Airport may be required)- Porto, I would have thought that Madrid, London Gatwick, and/or Paris CDG would be equally good routings and then catch a Ryanair or Easyjet low-cost to Porto.
From the airport you would have to take the Light Rail system into the city(Campanha) and then get a train to Braga. (Trains every hour) (About 50 minutes travel time)
