The only easy question to answer is: "How is the best way to get from city to city?" That's simple, take the Eurostar train. Less than 3 hours one way. It's the only way to go.
Have you been to either city? Three days it's enough at either. I've been to both cities numerous times and have the retirement goal of spending a summer in each. That said, you can still do a lot in your three days. While I love Normandy and the English countryside, I would suggest just staying in the cities because those day trips will take up the whole day and now your down to two days.
Get Rick Steves' London and Paris guide books and decide what you want to do.
Suggestions for London:
Westminster Abby
Churchill War Rooms
National Gallery
British Museum
British Library
Town of London
Lunch in a Pub (More than one every corner Steves' guide lists some famous ones)
Theater (I saw Wicked there last year and it plays in a large theater, I would suggest finding something in a smaller house)
Walking through the parks (Hyde, Green, St. James, etc)
Bike tour
That's well over 3 days' worth of stuff and I haven't mentioned a ton of great things.
Suggestions in Paris:
Norte Dame (Climb the bell tower)
St Chapelle (It's a must and most people miss it.)
Louvre
Musee d'Orsay
Arc de Triomphe
champs-lyses
Cruise on the Seine
Lunch on the west bank
Bike tour
Again, well over 3 days of stuff.
Like I said, pickup Steves' book and get a feel for what you're interested in.