Although London has all you could ever need, it's still great to get out of the city - daytrips by train are inexpensive if started after 10am and if you have a Network Southeast Card.
Bus services are also cheap and reliable - but are at the mercy of traffic.
Hiring a car is a viable proposition but parking can be a problem. Given to the public by Queen Victoria it boasts a succession of exquisite rooms, mediaeval kitchens, the famous maze, 'real tennis' courts and beautiful gardens. The palace is less than 5 minutes from Hampton Court station, which is 30 minutes from Waterloo. Worth half a day. Picnic in nearby Bushey Park.