American just back from two weeks and change in London ... with a side visit to Liverpool thrown in.
As someone else mentioned, the museums are your best value as most are free. The British Museum is so cram packed with stuff to gawp at that you could spend 2 full days there. The Victoria and Albert Museum is a favorite of mine ... more quirky and Victorian era specific, but terrific.
You mentioned the London Eye ... know that the tickets for this are stupid expensive, but it IS an awesome 30 minutes or so. The crowds will surge and shrink during the day for this, so don't be put off by a big line. Go have a pint or an ice cream and give it an hour.
Harrods is a particular highlight for my family as Harrod is our family name ... no connection obviously as the store is owned by the Fayeds. Still, it's a wonder of the modern world, that place. My tip, unless you REALLY want to pay London prices for a specific luxury item, is to pick up the store guide (yes, you'll probably need it to navigate) and head for the in-store gift shop. They have their Harrods branded keepsakes there. Then go to the Egyptian Escalator and find something yummy in the food halls and get out before you spend too much.
Oyster cards for Americans ... easy peasy, really. The street entrance level of any Underground or Overground station will have kiosks where you get get and load up an Oyster Card. We used the tube pretty much exclusively during our trip and budgeted 10 pounds a day per person. You can load that up and then check the balance as you go. Topping up is as easy as getting the card in the first place. Those cards get you on under and over ground trains and buses. Also note that there are several apps for Iphone or Android that feature London Tube maps and travel information. I fond one of those to be quite helpful. If you have time before your trip, you can go ahead and order Visitor Oyster Cards before you go. All the info is here ...
http://www.visitlondon.com/traveller-information/getting-around-london/oyster#k6GpxHzeUBypScl5.97
For budget purposes, know that the big ticket churches and the like charge 15-20 pounds per head with no family discounts if everyone is an adult. Westminster and St Pauls are worth every penny to me, but that's me. At that price do your own research ahead of time and see if it's worth it to you. Most big attractions have relatively informative websites these days.
Also ... EVERYthing is NYC level expensive and the exchange rate isn't helping matters. Remember that if you're on a budget.
We purchased the London Pass to help bypass some of the lines at popular locations and spread the cost over time, and we plan on getting the Oyster card for travel. I can't wait to go to Harrods! The food halls and the toy floor are 2 things I want to see. (Yes, I know...grown adult wanting to go to the toy floor. It's the only place I've seen a Steif on display! 
) The London Eye is for my kids. I'll hand off the camera and be the one at the bottom holding bags. 
We're from NJ so we know NYC expensive.... C'est la vie.
