Central London isn't that big, but there's no way you'll be able to cover all the main sights in one day, on foot. Not without knackering yourselves, anyway.
And buy an A-Z street map, which will help when travelling in London. Londoners use them too, so it won't mark you out as a tourist.
Get off the tube at Piccadilly Circus. Have a look around, then head east into Leicester Square. Keep going east into the Charing Cross Road, then turn down towards Trafalgar Square.
That'll take you down past the National Portrait Gallery, and bring you into the northern (pedestrianised) side of Trafalgar Square where you can go in the National Gallery, or take a look at Nelsons Column.
From Trafalgar Square, walk through Admiralty Arch and head down the Mall to Buckingham Palace. Head down Buckingham Gate and Buckingham Palace Road until you get to Victoria Station. Find Victoria Street, walk along it (possibly stopping to check out Westminster Cathedral - not the Abbey), and you'll come to Westminster and the Houses of Parliament.
You'll be within sight of the London Eye, which is on the other side of the Thames, but you can cross Westminster Bridge and walk along the embankment to reach it. From the Eye, it's about 10 minutes walk to Waterloo Station, from where you can pick up a tube, bus or taxi.
After that little jaunt, you should have had enough for one day.
Have a rest, then go back to Leicester Square in the evening to catch a movie.
I'd take a seperate day to do the City of London: Go to London Bridge on the tube, walk across London Bridge, go up the Monument (which commemorates the Great Fire of London - you can climb to the top), then head down King William Street to see the Bank of England and the Royal Exchange, before walking down Cheapside to get to St Paul's Cathedral (eleven quid each to go in!). Take a break, then get a 15 bus from outside St Pauls which will take you down to the Tower of London and London Bridge. Entery to The Tower was about 15 quid last time I went.