What a lot of people - even those of us who lived there - take a while to work out is that central London is actaully pretty small - the tube map is deceptive. You can actually walk a lot of it. As a result, I've got a 'tour' that combines walking and the tube and covers a 'day in London' with some of the major sites. Of course, each person is going to be different about what they want to see, but it gives you an idea of just what is possible on a Zone 1 ticket.
The first day is the prime 'walking day', but I'll include the rest as it shows how easy it is to get to many of the other attractions. Apologies for wording, I'm literally copying it from an email I wrote in May.
Day A - The "Walking day".
You'll start by tubing to Waterloo station from where you're staying. You want to do this quite early, as you'll hopefully have booked an early "flight" on the London Eye the previous night. At Waterloo station grab a breakfast (bagels etc) to go and look at the main train station part (where the overland trains go) (as seen in the Bourne Ultimatum and other movies).
Walk outside to the London Eye. Take your 'flight', try and stand river-side in the pod, but you'll get to move around in it anyway.
Leave the Eye, walk west (left) along the Thames to the first bridge, and cross it to St Stephen's Tower. (Big Ben is actually the bell, not the tower, and even that has just been renamed Elizabeth Tower). It's next to Parliament Square, which is handy. Walk around and get your photos. Across the square is Westminster Abbey, where many coronations/weddings/funerals have been. It's 18 quid, but spectacular and important.
From here, walk along Parliament St. You'll pass Downing St (Prime Minister's abode) on your left. You can't go down it, but there's police and security and stuff. Instead, keep walking until you see the Horse Guards Parade. Turn left and walk through it (it's free) to to the other side. You'll see some of the Royal Guard on horseback and on foot.
On the other side (where the beach volleyball was during the Olympics) you will see St James Park. Walk straight across it, and keep an eye out for the pigeon-eating pelicans (seriously!), and the many squirrels. At the other end of the park, you'll see Buckingham Palace. Hopefully you've arrived before 11.30am, which is time for the Changing of the Guard.
To the right of the palace as you look at it is Green Park. Walk through it to Green Park tube. A lot of this is where Paul did some filming. On Picadilly, turn right and walk down to Picadilly Circus. It's London's version of Times Square, in a way, but it's a common attraction to visit. Note the statue of Eros. Walk down Haymarket Street, and the tallest building on your left as you look up is NZ House. Note the amazing penthouse at the top with the balcony all around - one of the tallest buildings in London and we get it!.
At the bottom of Haymarket you hit Pall Mall. Turn left and walk to Trafalgar Square. Scene of many movies, including the pigeon feeding in Mary Poppins (DON'T feed the pigeons, it's now illegal). The National Gallery is the big building at the back of the Square, and you get a great long view up the Mall from Trafalgar to Buckingham Palace, if you look through the street with the giant archway.
Opposite that is The Strand. Look at that street, but rather than wander there, we'll head to the river. Walk down Northumberland Street. At the bottom of this is the Thames River again. There's a pub in front of you (two actually) in boat form. Head for the one called Tattershall Castle if you're hungry, and grab lunch and a pint - you have a great view of Parliament, Big Ben and the Eye from there.
When you're done, walk east (left) along the river, past Embankment tube station. Along here, look out for Cleopatra's needle - one of the two taken from Egypt back in the day. You're walking along Victoria embankment here - much of which used to be river. Continue along the embankment all the way until you reach the Millenium Bridge. Built for the year 2000, they had to add stabilization when it was too wobbly when people walked on it(!). Walk onto the bridge a bit and look back, you get a great photo of St Paul's Cathedral. Take your photo, and note that opposite St Paul's is the Tate Modern (which you weren't fussed about, but it's there). Head to St Paul's. Check out the Crypt (more famous people buried there) and the whispering gallery - including the amazing view over London. If you DO go to the Tate Modern, go to the cafe on the 5th floor for a great view of London City.
From St Paul's, walk down Cannon Street, to Monument. The Monument to the Great 1666 Fire of London is here, and you can go up it if you wish. Tower Hill is also nearby, where they used to hang people, but there's not too much to see, unless you have lots of spare time.
Down the hill from Monument is London Bridge (the 11th), it's actually pretty boring as a bridge. But if you go stand on it, you'll get a great photo of Tower Bridge. You can do this later tho, as you'll now be heading further along the river to Tower of London. If you're doing that this day, you may be short on time, but it's there. If not, continue to Tower Bridge, and walk over the Bridge. Its new paint job should now be finished, so it should be looking pretty nice.
On the south side of the river, you should see a funny leaning building that looks like it's slipping. That's the city hall(!). Odd. Walk past this west back along the Thames. Walk all the way to London Bridge, and at this point you may want to stand on the bridge for a photo of Tower Bridge. Now if you're going to a movie in town, pop into London Bridge station, and catch the northern line up to Leceister Square for the movies. If you have time for a pint, go to Waxy O Conners, just near Leceister Square, and for dinner - Mr Kong's in China Town (1 block north of Leceister square - walk past the ice cream shop and Prince Charles cinema, and you'll pretty much walk into it).
This would be the 'museum day'. If you were going to leave Tower of London to do this day and not the walking day, you'd do it today. Starting off, the British Museum. Then take the tube from Kings Cross / Russell Square down to Tower Hill/Gateway, and walk to Tower of London (taking note to look for the old Roman walls outside the tower).
Time it so you can start your wandering with a Beefeater / yeoman tour - they'll tell you about the tower. Excellent, entertaining and included in the price. Then it's up to you to see everything else - but when the tour finishes, bolt fast across to the crown jewels so you beat the rest of your tour group to them and get more time to look.
After Tower of London, walk to Bank station, and get the DLR (Docklands Light Railway) to Greenwich to go see the Observatory. This train is overground (for most of the time) so you can see Canary Wharf (big banking district with skyscrapers), docklands, and the O2 Arena (Millenium Dome) that James Bond slid down in one of the Pierce Brosnan ones.
From Greenwich, walk to the Maritime museum / through the Naval College, and up the hill to the observatory. You'll get a great view of London (weather depending) and you can jump back and forth over east and west of the line. Yay!
From here, tube back to London Bridge and at 7.30pm you can see a play at the Globe. Mon/Tues/Wed it's between Tempest, Midsummer Nights and that Scottish Play (not necessarily in that order). Book online when you know it's going to be ok weather. I recommend standing tickets for the novelty, but if you get seats - rent a cushion. Seriously. You'll regret it if you don't.
Head to South Kensington on the tube, and see the Natural History museum, and Science museum. You could potentially combine this with British Museum if you have time, or Tower of London / Greenwich depending on the day and weather, but not all of the above. Science Museum may sound less fun than Natural History, but when they have some Apollo modules and other amazing craft (some of the planes are epic!) it's worth an hour or two.
Are you taking a Eurostar to Paris? In this case, take a late one, and use this day to either use as a reserve for bad weather on one of the other days, or go see some of the parks if you haven't seen them. Or pop up to Regents Canal, or Camden Markets.
Tattershall castle - pub on a boat. Need I say more? Great view, and on the Thames.
Waxy O Conners - Irish pub with a giant tree (artificial) and many nooks and crannies, between Picadilly and Leceister Square. Amazing. Get a Guinness.
Waxy's Little Sister - just around the corner, good value but only if bigger Waxy's is full.
De Hems - Belgian and Netherlands pub just next to Chinatown. Incredible beers, and you may have to trade in your shoe if you want to order certain drinks...
Chinatown - Mr Kong's.
Brick Lane - incredible, cheap, delicious, Indian. Make sure to bargain a hard bargain - don't let them get away without including a beer and poppadoms and dropping the price at least once!
Pub food - tattershall, or else some of the touristy English pubs around Trocadero/Picadilly are actually quite nice, if a bit expensive. Samuel Smith pubs are all dirt cheap for drinks. Wetherspoon pubs are also cheap, regular quality and a chain(!) - their curries are very good, as are their lamb shanks.
Markets - in London Bridge station there are actually some pretty decent little stalls, and outside on weekends is Borough Markets - 1000 years and still going strong!