A) Dover:
it is some distance away from London. You could go by train, bus (National Express), or rental car. In Dover, the one and only thing to see is the Castle. It's really good; from up there, you can just catch a glimpse of the White Cliffs (it is difficult to get a full view of them, as you are standing on top of them, you know). If you have a car at your disposal, you could visit a place called Samphire Hoe, just outside Dover. It is, for me, the best place to see a bit of the White Cliffs.
http://www.english-heritage.org.uk/daysout/properties/dover-castle/
http://www.nationalexpress.com/coach/index.cfm
http://www.samphirehoe.com/
If you do a one-day round-trip by car from London to Dover and back, you might want to consider seeing Canterbury as well.
Note: there are public Tourist Information Centres in all tourist towns in England. I've been to the TICs in Dover and Canterbury myself. They are sometimes overcrowded, but the staff is very helpful, and if you end up in a place and don't really know what to see there, this is the first point to turn to.
B) London:
first thing you need to do after your arrival is buy a good map (of both the city and the Underground lines), and a small but decent guidebook (if you don't already have one).
secondly, get your Underground ticket sorted out. As far as I understand the "new" system, you will each have to buy an "Oyster" card (3 pouunds deposit) and then digitally imprint it with the ticket that is most suitable for your needs. If your hotel is within the Underground zones 1 or 2, you should get a "7-day travel card" for zones 1&2 (almost all attractions are within those zones), because it is best value for money (around 26 pounds).
As for sightseeing, there is always enough to see in London.
Most importantly, do visit the British Museum, and the Victoria&Albert Museum.
The V&A is directly next to many other museums, like the Science Museum and the Natural History Museum, so you might want to see them as well. Close by are the Royal Albert Hall and the Albert Memorial, which are well worth taking pictures of. At the Royal Albert Hall, they will currently be in the midst of their annual BBC Promenade Concerts season. Check out their season programme, and if their is anything on their that suits your taste, ask for cheap tickets (standing only places, for example), because the Hall is interesting from the inside as well.
If you are interested in paintings, the National Gallery and Tate Britain are the most famous places to go.
The National Gallery is next to Trafalgar Square (with Nelson's Column), St.-Martin-in-the Fields, and the National Portrait Gallery.
If the weather is nice, do walk around London's famous parks, or see if you can get tickets for an open-air Shakespeare performance in Regent's Park.
Other traditional sightseeing walks:
Look at Buckingham Palace from the outside, walk up the Mall to Trafalgar Square [National Gallery, Nelson's Column], walk down Whitehall [Horseguards Parade; Downing Street] to Westminster [Westminster Abbey, Houses of Parliament, Big Ben, ferris wheel]. That should be 3 to 5 hours, if you just stand round and take pictures and don't go into any of the buildings.
If you have time left by then, get the underground from Westminster to Tower Hill, to see the Tower and Tower Bridge.
Short walk: look at St Paul's Cathedral, walk over the Millenium Bridge, look at the Shakespeare Globe and spend some time at the Tate Modern.
Of course, the Tower of London is something of a must, but it is VERY expensive.
At least one day should be allowed for shopping. Visit Harrods, Selfridge's and Hamley's.
Walk down Oxford Street, London's famous shopping street.
All that should easily fill a week; if you still have time on your hands, you can do daytrips to Kew Gardens, Hampton Court, or Windsor Castle.
Avoid expensive and un-historical places, like Madame Tussauds, 221b Baker Street, and London Dungeons. They are basically not much more than tourist traps.
For the evenings: BBC Promenade Concerts at the Royal Albert Hall. Open-air Shakepeare in Regent's Park. Do get an edition of the Evening Standard newspaper daily. It has extensive listings of all theatre plays, musicals, concerts, etc. for that evening.