Cici,
Congrats on your trip.
These are the major things we would like to see/do: London Eye, Tower of London, Tower Bridge, Westminster Abbey, St. Paul's Cathedral, the Changing of the Guard at Buckingham Palace, Victoria & Albert Museum, Hyde Park & Kensington Gardens, Big Ben, Harrods, Windsor Castle, White Cliffs of Dover and possibly the National Gallery as well as enjoy a high tea somewhere and a ride down the Thames. We will also be during Trooping the Colour (not sure right now if this is a good thing or not!)...
Is this doable in the seven days we will have there (this time does not include travel days)?
I went in March, so im not sure about the crowds in June, but i will give my experience.
I went to the:
London Eye
Tower Of London
St. Pauls
Rode in Hyde Park
National Gallery
Plus others
I did my thing in 4 full days with out (w/o) any side trips. I didnt make it into Westminster Abbey (WMA) since it was closed. I did the the changing of the Horse Guards though since that appeals to me more. Instead of Harrods i went to Hamleys - at least i could afford something there.
With regards to the things you want to do, the National Gallery can eat up alot of time - in my case 1/2+ day. But in my opinion (imo) i think its worth it.
I should note that:
i had reserved tickets for the London Eye. No line for me, but long line for people that didnt.
No line for the Tower of London except for the Crown Jewels. If you go, do the Crown Jewels first.
Ive seen the white cliffs of DOver in passing, and imo, there isnt much, but if thats what you want, go for it.
Are there must-dos we are missing?
There are alot of museums and other things in London, if you can say what floats your boat others can chime in for suggestions.
We plan to use the tube and/or walk vs renting a car. I believe I read somewhere on here that we should purchase an Oyster Card....is this the best way to get around?
I personally use the Oyster card since for me, its a good deal and i use it when i go back. you will have to do the math on this one. I believe that there are other cards/passes that have better benefits for couples or muliples but will others more experience chime in on that. I love the Tube system in London and sort of ignore the buses and i shouldnt. I will try to use the buses more this year.
Also, what is the best way to get from Gatwick to London--Gatwick Express? If we decide to check out of our hotel and do Stonehenge/Bath at the end of our trip do we need to come back to London and then head to the Gatwick Airport or are there other options?
So far ive only used the Gatwick Express (GE). But there is the commuter train too. But this will depend on WHEN you want to go. Chances are its in the day and not in the early/late hours, so you should be fine.
As far as where you fly in/out of thats your business. you dont say where you fly in/out of.
just a general comment. some of these questions are answered in Rick Steves (RS) London book. I dont make any profit or gain anything by pushing his books, but my first time in Europe, i read his "Europe through the back door" and any of the city books (including London & Paris) he had at the time. I would say his books were 99% spot on. Another thing was that his book covers things to do/see and how to get there if they are outside of town or on "day trips". It takes alot of guesswork out of it. The only drawback on some of the books is that they dont give an estimate on how long it takes to go through the museum. I know that duration can be subject, but an estimate can be helpful in planning how much time to spend.
I dont strictly use RS books, but i use whatever i can get my hands on includng the www. I will google for things to do and see where im going just in case something that isnt as big a draw but appeals to me. YOU have the www at your fingertips, use it!
Happy trails.