Friends of ours - on their first and only trip to London - had one day on their own, scheduled by the package tour company.  On their 'free day', we suggested that they visit St Paul's Cathedral and the Tower of London.  They met another couple on the tour bus who invited them to spend the day with them... boat ride up the Thames to Greenwich and back. Our obliging friends went along.  At the end of the day, they asked the other couple why they did not want to see St Paul's and the Tower?  Their reply, 'Oh!  We did that on our first trip to London!'

Have a plan and stick to it.  That's Part A.  Part B is here.

While you are still in the planning stage, see our list of Favourite Web Sites - especially the 'Before you leave' section - and the Planning Tips and Day Trips from London pages.