one hour to beach:
Brighton, Britain's coolest Victorian seaside town, is 1hour 10 mins away on a direct train and makes for an easy day trip. Brighton has a great beach within walking distance of Brighton station, fantastic boutique shops in 'the Lanes', a huge array of restaurants and of course its vibrant night life. There is also a pier with an amusement park and the Royal Pavillion, King George IV's unusual palace, beautifully restored.
http://www.royalpavilion.org.uk/
20 mins to Wimbledon for tennis:
Direct train in 20 mins to Wimbledon. Ideal for the Tennis or a walk on Wimbledon Common. (look out for the wombles!)
English countryside and towns:
All within a couple of hours drive (or on the train) the following are accessible:
New Forest with its wild ponies, deer and stunning scenery. You could hire a bike and go cycling for the day in this unique National Park
Kent - known as the garden of England - with the infamous 'Leeds Castle' makes for a great historic day out.
The North Downs - beautiful green countryside with rolling hills
Canterbury - this historic city has a beautiful cathedral
There are many places on the South coast which have good beaches, watersports, etc which are accessible for a daytrip.
The stunning roman spa town of Bath with its newly refurbished Roman baths and georgian architecture is just an hour and a half by train from London.
For a city of culture and football Manchester is around 2 hours 30 mins on the train from central London.
On the Eurostar it's 2hrs 15 mins to Paris. Brussels and Disneyland Paris are other potential daytrips.