With rail, it doesn't take long to get anywhere.
...and you will only pay an arm and a leg, too!
I try to stay as far away from London as possible so I can't be much help I'm afraid. What I will say is prepare to pay quite a bit for lodgings because everything in London costs more than the rest of the country. A phenomenon known as London weighting. People there even get paid higher salaries because of it. Also, if you're planning to drive around you should be aware of the "congestion charges."
As far as things to do, don't forget London Zoo. I was underwhelmed by it but most people seem to love it. Use the Tube to get around, but have at least a glance at the website/map first so you know what to expect. I know some
people who live in London so I'll ask them for their thoughts.