The route I would take to County Hall would be tube from Heathrow to Westminster, as Bard suggests, which will mean a change, but my tip is to change at Barons Court, where you simply cross to the other side of the platform for the District Line (the other changes mean stairs !!). Then it is a lovely walk across the bridge with the Houses of P on one side, the London Eye spinning away on your left, etc etc. Some of the best tourist bits as soon as you arrive. The A2 bus is more expensive, less picturesque (discounting the part of the tube actually underground) and less representative of life in London.

As to getting around in London, the Tube again is hard to beat for most trips - the buses are excellent, but take some learning. But best of all is shoeleather, inexpensive (unless you habitually wear Bally shoes) and the most productive method. London is largely about details, once the big edifices are out of the way. Several strolls thorough Hyde Park & Kens Gardens are essential, closely followed by St James' Park, with Green Park on the way.

One tip though, assuming you are not already well acquainted with London, is to take the Red bus from Park Lane (or any of its stops) for a day. Used to be 10pounds, probably 15 now, and it takes in all the sights of The City and south bank, off and on as often as you like, for 24hours, all with commentary. There are maroon and gold buses and possibly other ones too, but all much the same value, and all terminating just near Speakers Corner.

While you are near Speakers Corner, take a wander around the Marble Arch, and reflect on the ghosts of the people executed at Tyburn, the gallows of which were on the site of the traffic island at the corner of Edgeware Road.

Sorry to ramble on, but you've hit one of my favourite subjects.

Paul in Rainy Sydney