One of the exits of Oxford Circus (Central, Victoria, Bakerloo lines) is Argyll Street which is pretty active, sometimes has street entertainers, and some interesting architecture in the area. Although I have not taken advantage of it myself, it always seems to me to be a pretty photogenic spot for street photography.
It also leads into Carnaby Street, famous in the 60s for its fashion and music stores but these days is mostly the usual chain stores, while Oxford Circus is at the junction of Oxford Street and Regent Street, London's main shopping streets. Oxford Street is the more main stream, Regent Street has the better architecture.
The actual City of London itself is pretty small, famously being one square mile, and so easy to walk around, Bank station (Central, Northern, Waterloo & City, DLR) sits at its centre. Though like most financial districts unless you want to see bankers or buildings there is not much else going on. Near Monument station (District, Circle) is the monument to the great fire of London, and has a viewing platform at the top which is only a few pounds and is supposed to have great views, though it means climbing a 200 foot staircase! Bank and Monument stations are linked.
The monument sits on the southern end of London Bridge, itself not very interesting but as the next one along from Tower Bridge it gives great views to one of London's most photographed attractions. It also gives access for walking up and down the river.
Also in the city to the East of Bank and the monument is the Lloyd's Building on Lime Street, which was designed 'inside-out' so all the stairwells, conduits etc. are on the outside making the space better inside the building. And there is also on of London's newest towers, the Swiss Re Building on St Mary Axe, aka the Gherkin.
Heading West from Bank takes you St Pauls' Cathedral, and following Ludgate Hill from there it turns into Fleet Street, formally the main location of Britain's press, and at the end where it becomes the Strand sits the Royal Courts of Justice, home to the highest courts in the country. As well as the chance to see barristers (lawyers) and judges running around in robes and wigs, the courts are open to the public.
Personally I find Soho to be a bit boring (great for vegan food though!), this is the area borded by Regent Street, Oxford Street, Shaftesbury Avenue and Charing Cross Road. Piccadilly Circus (Piccadilly, Bakerloo) though, at the Junction of Shaftesbury Avenue and Regent Street, is interesting and Coventry Street leads from there to Leicester Square which is where the London movie premieres take place. And immediately to the north of that is Chinatown. You could also use Leicester Square station (Piccadilly, Northern [Charing Cross branch]). Heading east from here leads to Covent Garden which is home to a craft market and usually a lot of street entertainers. While it has its own station (Piccadilly), it is small and often closes due to overcrowding as platform access is only by lift or a spiral staircase.
To the south of Leicester Square you come to Trafalgar Square, home to Nelson's Column, the National Gallery and National Portrait Gallery (some exhibitions are charged, but there are free photographic exhibits too). Nelson's Column sits at the top of Whitehall which houses many government buildings, including Downing Street, the office of the Prime Minister, protected by a large fence and armed officers, before culminating in Parliament Square, and the houses of Parliament. Turn onto Westminster Bridge and you come to face London's other most photographed attraction, the London Eye.
I am going to stop now as I am beginning to sound like I work for the tourist board! The above (well maybe except the city) are the 'obvious' tourist locations to see and do, and for good reason, but most of central London has something to offer. If you want to be more more unique just get off of a train anywhere and just walk around and you will find quiet squares, bustling streets, etc. Roughly speaking, the east side of zone one is the city, the west side are the more affluent neighbourhoods, while between them sits the west end.
If you want to see bridges lit up at night two of my favourites are Albert Bridge and Chelsea Bridge, both suspension bridges and so rather different from the usual brick and stone ones that most people are used to further downstream. The former is pink and white and with the lighting feels more Parisian, while from the latter is less ornate and from closed up has a more colder raw metallic feel. There are also a large number of moorings around the north side of Albert Bridge.
On the south bank Battersea Park sites between the two, which is a nice park and features a large Pagoda. If you start here and head west to cross the Battersea Bridge, walk along Chelsea Embankment towards Chelsea Bridge. Staying on the north side if you go just a little way beyond the bridge down Grosvenor Road (which Chelsea Embankment becomes) and under the railway bridge you are directly across the river from, and with unobstructed views to, the iconic Battersea Power Station.
Finally, if you are into people watching and interesting neighbourhoods I would also recommend Camden (Camden Town station on the Northern line) though this is in zone two just north of Euston and Kings Cross station. Primrose Hill (west of Camden, and across the road from Regents Park) gives decent views across London, while the canal from Regent's Park goes through the centre of Camden and towards east London which is interesting to walk along.
