I hate to sound like your mother but if you're going in June, you should have had this all nailed down by now.
A bunch of random thoughts:
Anyway, I like the half in London half in Paris idea. If it's your first time going to Europe, this is a great way to do it. Eurostar between the 2 cities is the only way to go. I also like the idea of getting Rick Steves' guide books for Paris and London. He'll give you the basics on what to expect and where's a good place to stay. You'll need to learn how to use the Tube and Metro in each city. If you haven't used a subway before, don't worry, they're pretty easy. Again, Steves will explain all this. Get an Oyster card in London to save money and the Paris Metro allows you to buy tickets in bulk to save a little cash. There are pick-pockets on the trains and in all popular locations, again, don't worry too much, just be smart. Get a money belt that goes inside your clothes and avoid crowds (if possible). I have a golf windbreaker with a zippered pocket on the inside. I keep my cash and credit cards in that pocket. Before you leave, go to your bank and get several 100 Euro and Pounds for initial walking around money. Get smaller notes (i.e. 5s, 10s and some 20s). Ask your bank about using ATMs in Paris or London. That should be easy. I like to put most of my charges on my credit card. Visa and Master Card are excepted pretty much everywhere. Nowadays, they may ask you if you want your bill charged in the native currency or in US dollars. You should always charge in native currency, they generally give you a bad exchange rate for the privilege to use USDs. Let the credit card take care of the exchange rate. One last thing about credit cards, the bank that owns your card may charge you a fee to use it in Europe (usually about 3%). Call and ask. Several years ago, I switched to a Venture One card that has no fee. It was the one Rick Steves' blog suggested. Your credit card will need a chip to work in Europe but I think now all/most credit cards have one so that should not be a problem. Another thing to worry about before you leave is international data rates for your phones. Call your provider and make sure you've got it covered. I've heard horror stories of people screwing that up and getting $1000s in fees.
London:
Must sees: Westminster Abbey, Tower of London, British Museum, National Gallery, Churchill's War Rooms, St. Paul
Must dos: Take one of these bike tours, go to a pub for lunch, see a west end show, London walks tours, visit Hyde Park's speaker's corner on a Sunday, Thames river cruise, walk Whitehall from Parliament to Trafalgar Square then down the Mall to Buckingham Palace, be in Parliament square when Big Ben chimes at the hour, take the Millennium bridge from St. Paul's to the Tate Modern.
Day Trips: Hampton court is nice. I think a first timer to England should take one to Stonehenge and Bath. This will give you a good tour for English country side and a nice English town. There are tour buses doing this, look at trip advisory.
Paris:
Must sees: Louvre, Musee d'Orsay, Notre Dame, Saint Chapelle (many miss this but don't), Sacre-Coeur, Eiffel Tower, Arc de Triomphe
Must Do: Bike tour, Find a caf on the left bank, walk from the Louvre to the Place de la Concorde (find the guy juggling pigeons), walk the Champs-Elysees, Seine river cruise at night, beer at Montmartre
Day Trips: Versailles (easy train ride), Normandy (I love Normandy but if you try this remember it's 3 or 4 hours each way depending on traffic. Again, you can find these via trip advisory. If you're a D-Day fan and want to do this, it will be very rushed but at least you'll see Point-du-Hoc, the American Cemetery and Omaha Beach), Giverny (Claude Monet's garden with all the water lilies. It's maybe an hour and a half from Paris, should be able to find a tour on trip advisory)
London and Paris are two of the greatest city in the world. Have fun and soak up a bunch of the culture. Walk around and just wonder into a pub or caf. That's where you find some of your best memories.