From LHR: Personally I would recommend the HEX to Paddington and the Bakerloo line to Charing Cross. It's quick. However, it's expensive - £13.50 for the HEX and £1.50 for the tube if you put it on the Oystecard (£3 deposit for the card or £3 for the tube (old-style paper ticket). If cost is an issue, the take the tube (direct from Heathrow 1/2/3) - £3.50 with an Oystercard, £4 without.

Covent Garden is a direct ride from LHR, and is very deep with a few steps from the platform to the lifts, and lifts to street level.

From LGW I would take the Thamelink trains to Blackfriars, which is direct, runs every 15 mins (trains marked with destination Bedford) and gets to Blackfrairs in 25-40 minutes. From Blackfriars it is a short walk along the embankment to Somerset House, or one tube stop to Temple.