Things to see in London...
Well, the trouble is, you see that there is soooo much. I would recommend:
The Museum of London - small museum within walking distance of Moorgate tube station (you can take the sky walk from Moorgate all the way to the museum, past Barbican). Includes interesting potted history of London from stone age to today, often has interesting small exhibitions from the collections.
The Tower of London/Tower Bridge/HMS Belfast are all within walking distance of each other - from Tower Hill tube station. About fifteen minutes walk from there is Petticoat Lane, which has a busy street market every day. About ten minute's walk from Tower Hill is Leadenhall Market, which Harry Potter fans will recognise...when they go to Diagon Alley in the films, they always seem to set off from Leadenhall Market. Don't expect good shopping there though - it is mostly food, game and cheese etc. There is a Chinese alley around there which is good for obscure Chinese culinary ingredients

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There is a huge amount of stuff within the square mile of the city. The Guildhall has a permanent exhibition about Roman London I believe, although it is years since I went there. There are bits of Roman wall visible from the Museum of London, but in many other places too.
For shopping, people usually talk about Regents Street and Oxford Street, or if rich, Bond Street etc. The locals go to Camden to shop...and Camden Market. It's very "in" at the moment, lots of boho fashion and small indoor markets, handmade and imported stuff. It is very busy and grubby, but the bit around Camden Lock is interesting too.
Covent Garden is also good for shopping - but I would take the tube to Holborn (pronounced Hoebun) and walk from there rather than taking Covent Garden tube - it is very deep, and waiting for the lifts is a pain. There are good places to eat in the area around Covent Garden, in the market area itself (which used to be the old fruit, flowers and vegetable market) places are expensive, a walk around may find better and cheaper. There are market stalls and lots of street theatre, mime, jugglers etc here.
For cheap places to eat you can't beat the Chinese restaurants around Leicester Square...but be very careful in the area of Leicester Square - it is a haunt for pickpockets, and people trying to part tourists from their money. Some years ago I was pressed to take a job by a seedy looking man who offered to lend me 10p to use the phone. As I was working as an editor at the time, I declined his offer of 10p and the job, but I reckon it would have been hard and horizontal work

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There are many galleries in London - there are galleries large and small. I like Tate Modern and Tate Britain, which are now linked to each other by a fast river bus - I don't know how much this costs, however

. The river is a good way to get around the main sites. If you go to the London Eye, you could take the river bus from the embankment on the opposite side of the Thames (over Westminster Bridge, past Big Ben and all that stuff) and go to the Tower of London by river bus. Some of the new Catamarans are fast and smooth...some of the gronchy old river boats are tossed around so much that it seems a miracle when you arrive the right way up.
There is a lot more to do and see that I can ever include in this post...there are sightseeing buses, but these tend to be expensive, whereas an all zones ticket for the tube includes travel on the good old London Buses...no tour guide, but you can see as much from the top of a London bus as a sightseeing bus, for nothing.
The British Museum is wonderful for anyone who likes history, architecture, anthropology. I'm afraid we nicked a lot of this stuff from the rest of the world, but we have looked after it

. If you like museums, there is enough there for a whole week....
There are also several large museums in South Kensington - the Natural History, Science and Victoria and Albert Museums are all clustered here.
Not knowing what your interests are...that's about all I can think of for the moment, but if you have a particular sorts of things you enjoy, post them and I will do my best to come up with ideas.
best wishes
Cali