From London LHR there are flights to Madrid daily at 0725, 1215, 1525, and (every day except Monday) 1635 which now depart from Terminal 3. There are also 8 BA codeshares from LHR and 3 from LGW on which you get access to BA's free bar in economy, which can provide a much better ride if you are in this class.
Iberia MD87s at Madrid
There are also daily Iberia flights from LHR to Barcelona el Prat at 0620, 1125, 1630 and 1915. There are also 4 BA codeshares from LHR and 2 from LGW). Indeed, there is a small Iberia hub at Barcelona, which the airline has long sought to turn into a hub. There are new long haul routes to Miami and Sao Paulo.
To Valencia there is one daily flight from LHR, at 1715, which is generally scheduled too late to make any useful onward connections. Seville also gets a daily flight at 1110 - in an MD88, as does Bilbao at 1740.
Iberia has its main hub at Madrid. There are international flights to America with destinations including New York (A340s daily at 1305 and 1700, Chicago (daily except Monday with an A340 at 1200 and 1425, Boston (six flights a week in summer), and Miami (daily at 1205). However, the very poorly patronised route to Washington has been axed.
Iberia 757 at LHR
There are also A340 services to Mexico city, Havana, Guatemala, San Jose, Panama, Santo Domingo, Caracas, Bogota, Lima, Santiago, Buenos Aires, Montevideo, San Paulo, and Rio Janeiro.
To Africa there are flights to Dakar, Lagos, Malabo, Johannesburg, Nairobi, and Cairo. Further east you can get to Tel Aviv, Athens, Istanbul, and Dubai.
Around Europe Iberia fly to destinations including Dublin, Paris, Brussels, Amsterdam, Dusseldorf, Fankfurt, Lyon, Nice, Zurich, Milan, Rome, Munich, Stuttgart, Hanover, Berlin, Copenhagen, Stockholm, and Vienna.