OK, well if you're going to drive in, I'd advise choosing your approach carefully, and avoiding the busier times (6am-10am, 4pm-7pm weekdays). The A20 is a good approach, as you've got a long stretch of dual carriageway, but it ends abruptly at the roundabout with the A205 South Circular at Eltham.

From there, it's a fairly straight shot on surface streets through Lewisham, New Cross, Camberwell, Vauxhall and Victoria on the A20/A2/A202 to Hyde Park Corner, which will take you up to Piccadilly.

For a less simple/direct but more sightseeingy route, carry on from Eltham up the A20 to the A2, then follow the A2 over Tower Bridge. Turn left directly afterwards, and head along Lower Thames Street towards the Embankment and Westminster - you'll see all sort along the way. From Westminster, you can cross Hyde Park for Hyde Park Corner and then Piccadilly (or you can come off at Northumberland Avenue and go round Trafalgar Square).

Don't expect anything but surface streets for the last 10 miles, and expect the traffic to be slow-moving for that portion (and in the approach to the roundabout at the end of the A20). Make sure you know the rules for roundabouts, mini-roundabouts, and watch out for London's famous "gyratory systems" - big and often baffling, circular one-way systems with lanes that branch off in different directions, that often require you to cross several lanes of moving traffic to reach your destination - where the directions are often marked on the road itself rather than on signs. I have to cycle around two of these every day and they are not for the faint-hearted! But do use your indicators (signals) when you reach these - London drivers are generally quite forgiving, as they know how hard it can be to cross lanes when you make a mistake. Expect a flash of the lights to indicate someone letting you go.

Personally, even in a group, I rarely drive into London from where I live - only when I know I'm going to be staying out past the last train (which is about 1am - so that's a big night out!) and know I need to be back home. It usually ends up being rather boring and stressful. The noble exception is when I'm doing the Christmas Day shuffle, in which case it's great fun to drive through a deserted London past all the sights.

Under no circumstances use the A23 to drive in. It's an awful, awful road.