From Reddit London wiki
- Prehistoric stuff: well, maybe not prehistoric, but you can still see bits of the "London Wall" around the City of London area (Moorgate, Barbican) which is pretty cool (and free!). The Museum of London (also in that area) is free and has some other cool old stuff from the city's earliest history, including the original gate to the notorious Newgate Prison and an actual cell from inside it. Also in that area is the "London Stone" which is an ancient marker used by various tribes to establish rule and make laws upon -- it's embedded randomly in a wall somewhere in the City.
- If you head to the museums district in South Kensington you can see the Natural History Museum, the Science Museum, and the Victoria and Albert ("V&A") one after the other. In the case of the latter, they have this amazing room called the Cast Courts full of copied full-scale models of huge classical works (statues, columns, sculptures etc). Most of the things they copied are now destroyed so the copies themselves are the only opportunity to see how big things like Trajan's Column were in real life. The models are from the 19th century so they're pretty old in themselves -- worth checking out.
- The Gladstone (aka "The Glad") in Borough (just south of London Bridge) puts on free music every weekend and it's really good. It's mostly alt.country / folk and occasionally classic rock'n'roll, and since it's super small it always has a really intimate and warm atmosphere. Also has good beer on tap.
- Waterloo graffitti tunnel, under Waterloo Station. It's a pedestrian underpass absolutely covered in in graffiti. If you ever come back to London, you could be sure it would change too, so you could return to find new stuff.
- John Soane Museum. Not been, but heard very good things. http://www.soane.org/
- Check out the Barbican, because it's cool to look at, but also see what's on there. Often have shows or exhibitions, sounds like probably your cup of tea. The Barbican is London's tallest residential blocks. People either love them or hate the huge brutalist buildings and the area feels like a strange post apocalyptic world. The centre is used for arts. http://www.barbican.org.uk/
- Chistlehurst Caves. Perhaps not so much your kind of thing, but it is the most unusual attraction in London. Man made mines carves by Druids, saxons and Romans, used as air raid shelters in the wars, in the 60's David Bowie, Status Quo, Jimi Hendrix, The Rolling Stones and Pink Floyd played down there, art carved into the chalk walls, Cradle of Filth filmed a video there, people do LARP there... Okay, perhaps some of that doesn't sell it, but it's interesting and rarely busy and costs £5 for a 40 minute guided tour of awesome. I recommend greatly, unless you don't like the dark. Most Londoners don't even know about it. The website won't help sell it, but it's seriously worth checking out.
- Two arty museums definitely worth a look: The Huntarian Museum and The Wellcome Collection. They're pretty near each other - the Wellcome is opposite Euston Station and The Royal College of Surgeons is further down the road in Russell Square. Just around the corner from the British Museum if you're having a heavy museum day.
- Start off by visiting Highgate Cemetery. It's an historic cemetery full of many famous burials and massively interesting sculpture set in gorgeous woodland. I've linked to the page that has tour information - it's well worth it.
- Head up the hill into Highgate itself and get yourself some lunch at one of the pubs up there. The Angel does a good roast & has a good selection of british ale and european weiss bier.
- Wander through Highgate (it's a lovely area full of interesting houses) to the north of Hampstead Heath. Walk off that big lunch.
- Have a gander round the Heath (great views over London, a lovely open green space), then head over to Kenwood House. That link's currently broken, but hopefully it'll work by the time you come to check it out. The House is a gorgeous old stately home full of art and old paintings of the area as it was at the time. It's a great way to spend a couple of hours, if you like that sort of thing.
- Wander back down through the Heath towards Hampstead to find yourself somewhere for dinner. There are a bunch of restaurants and pubs with good food around there, but my choice would be The Garden Gate.
- Finally (yes, all my favourite places are pubs), head back past the Heath to the east and find The Southampton Arms. It's my absolute favourite pub, and it's a great place to spend an evening.
- You could always look for historic pubs like these. (That article is from 2006 so check for their current status first). Another list here. Alcohol and history, no need to chose.
- I suggested this templar church to someone else here recently. Genuine crusader tombs with the dog at the feet. I think it's associated with the Da Vinci Code but it's still wonderful.
- Try a river boat trip, it's an interesting view of the city and you can get boats from very near the houses of parliament. You could also consider the boat trip to Hampton Court but it takes a long time so going both ways is hard to time, and leaves with not a lot of time there.
- Museum of London will have the most local history information.
- Try visiting the botanical gardens at Kew. Our botanical gardens come with extra history and weird royal connections (one odd historic fact - there were workers from Kew on both sides of the mutiny of the Bounty). It's near Richmond where there should be some nice pubs overlooking the river.
- Forget london tour buses. Go to Aldwych or the Strand and wait for a very old looking number 15 bus. This is an original routemaster and if you have a travelcard it's free, if you have an oyster it's just a normal bus. It'll take you to tower hill (where the Tower of London is) and past St. Pauls and heaps of places on the way. The bus conductor will be happy to talk to you, give you some ideas and take photos of you if you want.
- Get off at St. Pauls and do that, it's a 400 year old cathedral slap bang in the middle of London just on the edge of the City. Walk down Ludgate hill when you're done, cross over and keep on the right hand pavement until you see a dodgy old looking building. This is ye olde Cheshire Cheese, it was Charles Dickens' local pub and dates back to 1667, although it was built on the remains of another pub destroyed a year earlier in the great fire of London. Incidentally it's a Sam Smiths pub, so the beer will be extremely cheap for London. Don't forget to check out the guestbooks signed by various people in history.
You need to decide whether or not you want to do the tower of london. It's pretty expensive and most Londoners will tell you to avoid it but it has got several palaces, you can easily spend most of a day there and it's rammed full of history.
- Don't bother with Buckingham Palace, you won't get in. Only do it if you're going to Hyde Park or walking from there to Westminster (which is a nice walk on a sunny day), Big Ben and Westminster Abbey.
Covent Garden's quite good for nights out. Book the Great Queen St. (do it now, you won't be able to turn up on the night), expect to pay between £30-£50 per head for the most amazing meal and wine together you could have. A little down the road you have Lowlander, which is good for a belgian beer (or several hundred), Belgo which is similar or the Porterhouse, which is a Dublin-based pub that has beers from all around the world.
- If you're into Sherlock Holmes, then his house is on Baker St. Of course it's not his real house but it is meticulously decorated. Again, Londoners would say it's a rip off but if you want something typically British to show your friends pictures of after they refuse to believe you went to a fictional victorian detective's apartment, this is it.
- Not too far away from Paddington is a great breakfast place called Core, it's on the waterfront. You can walk from Core to Paddington, take a left at the crossroads beforehand, head down the street and at the bottom you'll find an old Gin palace called the Victoria. As gaudy as it looks in the bar, head up to the top floor for a secret. It's about a 20 minute walk from there to Buckingham Palace and then you can do the walk to Westminster.
- In Westminster you have the houses of parliament, Big Ben and Westminster Abbey. If you go across the bridge, make sure you turn around to take a photo. Turning left at the end of the bridge will take you towards waterloo and the London Eye. You can walk a fair bit along the waterfront for some great views (views from the south bank are better than from the north because of planning rules around St. Pauls).
- If it's raining go to the museum district. The V&A is free and is a must see, as is the Natural History Museum, which may still have it's Darwin exhibition on. It's also worth checking out the British museum, which for it's name has a remarkable amount of paraphernalia pilfered, pillaged and plundered from around the world.
- Another must see (if you've never tried it) is the BFI Imax near Waterloo. The screen is 3 stories high and the films are in 3D.
- If you want to give it a try, go to the Greenwich Meridian Line and you can stand on the exact point where mid-day is actually mid-day according to GMT. You're about as far from the International Date Line as you're ever going to get. Several seconds after that's done, go to the Royal Observatory and check out the Planetarium. Make sure you take the DLR either there or back so you can pretend you're in Blade Runner with the tall buildings around Canary Wharf and the overground rail going across the water. Ok, maybe that's just me. I'll shut up now.
- Do take the bus and try to walk during the day, you'll see so much more of London. Part of the must see stuff is because it's off the beaten track. It's the little things like walking past Karl Marx's house, or checking out some of the places Dickens used to hang out in when he lived in Bloomsbury or spotting various celebs on the Kings Road in Chelsea that will make your experience memorable (although I've just noticed you live in LA, so maybe scratch the last one).
- If the weather's good and you're a bit bored of London, take the train to Kew, do Kew Gardens, then hop back on the district line to Richmond. It's a quaint Surrey town on the edge of London that has a beautiful park, a very traditional cricket green, tea rooms and if you head to the waterfront and turn left (then keep going) a really lovely outdoor Bavarian waterfront restaurant/bierhaus that's not too expensive but is delish.