<font face="Verdana, Arial, Helvetica, sans-serif" size="2">Originally posted by stut:
If cost is an issue, you can take the Piccadilly Line from Heathrow, changing at Holborn for the Central Line.
From LGW, hmm. Probably the easiest is to catch the Thameslink to Kings Cross, and get the Circle/Met/H&C from there. Otherwise, you could get the Thameslink to London Bridge and cab or bus it.</font>
If you have luggage, want to use the Tube from Heathrow to keep the cost down but don't have time problems, I would actually recommend a two-change trip on the Tube. Piccadilly Line to Barons Court. Cross-platform change to District Line. Change again to the Circle Line at any station from South Kensington to Monument - same platform, just get off and get on again. The amount of physical humping of luggage up and down stairs and escalators, and along corridors, is kept to a minimum this way.
From Gatwick, I agree with Thameslink but would get off at Blackfriars and change to the Circle Line there. (Much longer walk from Gatwick Express to Tube at Victoria.)