Hello,
I am sure there will be more people here to help with this...
The best bank? It depends on what you want to do.
If you are down here running a small/medium business I would say stick with one of the Cajas (CajaMadrid, LaCaixa, CajaSol, etc) else I would stick to a bank which has a good number of ATMs and won't charge you monthly fees or money transfer fees (ING, OpenBank from Santander, IBanesto from Banesto, Bancaja, etc).
I know Barclays has offices in Madrid. But I have no idea how they operate.
Another factor is that banks are "stronger" in different areas of the country so it will depend on where you will be.
Cheers