I have not been on an International flight in quite some years, and I would like to at least pick a quality airline that is not ridiculously expensive.
I'd fly Virgin Atlantic from Miami to London, then Iberia from London to Barcelona. Virgin Atlantic will be about the cheapest, and on par (in cost) with Lufthansa, United, Alitalia (never fly with them!), Air France and BA. Virgin Atlantic's service is bar none, they give you all sorts of free stuff and have great on board entertainment. For more info, see:
Iberia if you book at the right time will be pretty cheap ($20 e/w + tax) from London to Barcelona. Use http://www.zuji.com.sg 's flexible date feature to find when the cheap flights are and plan around that.
I'm not sure if it is expensive for you or compared to the other international airlines, but personally, I think Singapore Airlines is good in terms of quality and service.