I would definitely say check out the Wikitravel page for London if you need some general information (especially in regards to public transport). I think they give a fairly good overview of the famous sites in London:
Free walking tours are always a really good way to get to know a city. I just Googled “free walking tours London” and there are numerous tours out there. A friend has recommended SANDEMANs walking tours to me before in other cities so I reckon the London one must be good too.
They cover quite a lot of London like Big Ben and Houses of Parliament, Westminster Abbey, Buckingham Palace and all that.
Other sites worth checking out are definitely Tower of London and Tower Bridge, St Paul’s Cathedral (walk across the Millennium Bridge too). Leadenhall Market is also quite pretty and not too far from the Tower of London if you’re passing by.
If you get the chance and you enjoy markets, spend a day in Camden. You can catch the Tube to Camden Town and spend the day wandering the town and the markets there. You may find some really cool vintage finds!
I would recommend the National Gallery of London too. There are so many museums and galleries in London it’s really hard to pick one to visit. Victoria and Albert, British Museum, Tate Modern, National Portrait Gallery… I’d recommend at least one.