Step 1: Book a round-trip flight that lands in London early in the morning and takes off late at night (these flights are usually cheaper)
Step 2: Book a round-trip Eurostar ride to Paris on the internet (where there are usually better deals)
Step 3: Book your hotel rooms in advance online. Find the cheapest hotel, close to the major attractions, be open when you finally get to see it in person and be happy that your reservation actually exists and is accepted, no matter what state the room is in
Step 4: Use your feet and public transportation to get you where you want to go. It is the best way to experience the local culture and see its normal architecture. If you are really a chatty person, you can take a cab, but stay away if you don't plan on talking with the driver - you'll only be missing experiences)
Step 5: The Itinerary -
Day 1 -
Upon arrival in London, take the Tube to your cheap hotel that you'll undoubtedly regret booking and store your luggage at the front desk. Head out immediately (on foot, if possible) through Notting Hill to Kensington and Hyde Park. Take a nap on one of the park benches if necessary. When ready, continue on foot to Green Park and Buckingham Palace. You will most likely walk quickly because it is not a pretty area and you will arrive just in time to catch the mid-day Changing of the Guards. If you're lucky, you'll have a quick afternoon tea with the Queen and then be sent off again, this time to Westminster Abbey and the Big Ben. Note that there are two lines (queues) to enter the Abbey - one for tour groups and one for individuals. Check which line moves faster and join it. Finish the day with Fish and Chips to gain back all the calories you lost from walking.
Day 2 -
Take the first Eurostar from Waterloo Station to Paris. On the two-hour ride you will have to use your imagination to enjoy the view (after all, you will be riding through a long tunnel). Upon arrival, take the Metro to your 5-star hotel where the receptionist refuses to speak English and the bathroom is missing a light bulb, toilet paper and a wall. Leave your luggage and step out to explore the city. Begin at Notre Dame and walk along the river to Pont Neuf and to the Luxembourg Gardens, where you can take a quick and much needed nap and pay for using the dirty toilets. When ready, continue to the Pantheon and the Latin Quarter. Find a restaurant that serves escargots for dinner. When night falls, take a stroll to see the Eiffel Tower in the dark.
Day 3 -
Get up in time to catch the first elevator to the top of the Eiffel Tower, before all the crowds, and take many pictures (the view is nice and you have to prove to people that you paid to go up there). Descend by the same elevator (there is no time to bungee jump) and continue your tour to the Arc de Triumphe, les Champs Elysees, l'Obelisk and Tuileties Gardens.
Day 4 -
Le Louvre. Take as much time as you need. This is the only day allotted for this museum (keep in mind that it is said to take at least a month to see it all). If you are only there to say that you've been there, you can do it in five hours. Afterward get out and eat something. A baguette is a good option. Finish the afternoon is Montmartre and the Basilique de Sacre Coeur. Good place to buy a crepe, sit on the grass and enjoy the view that unfolds to the south.
Day 5 -
If your trip back to London is later in the day, get up and act like a Parisian: Start your day with a cafe and croissant and then head to nearest park with a book. Make your way back to London and your lovely hotel. The management will be so happy to see you have returned that they will upgrade your room to one that actually has flowing water and an unused soap. Take advantage of the situation.
Day 6 -
Do the things you missed on your first day: London Museum of Man, Tate Modern (one of the most interesting, especially because admission is free), the London Eye, London Tower and Greenwich (not the most interesting place, but at least you can say you've been to GMT-0 and floated on the river). Finish with Piccadilly Circus. This is a great place to grab a bite to eat and drown in the crowds for the authentic London experience before leaving.
Have a nice and safe trip!