London native here.
The Shard. Start with the Shard. It'll cost you about £28, but you get as much time as you like looking at the world's best city from floor 69 (haha). I've lived here for 15 years and when I went up a couple of days ago I still felt amazing. It's a fantastic starting place for a trip as well, you really get the lay of what is a mind-bendingly confusing city.
If you've never been to London before then it's probably worth going to all the normal places for souvenir value, but really the best part about London is the fact that there are so many places which are cheaper, less RAMMED FULL of tourists, and are deeply surreal to boot. Here's a list:
The Hunterian Museum
(Lincoln's Inn Fields, WC2)
Largest collection of jarred body parts. Probably. Part of the Royal College of Surgeons and a genuinely troubling museum experience.
Sir John Soane's Museum
(just opposite, in fact)
Stand-out exhibit is a jar of piss. It's completely packed with random stuff this one fella made or found or stole and brought back. Honestly one of my favourite places in London.
Not Camden
(anywhere but)
Oh dear. People have suggested Camden. Please don't go to Camden. I'm going to quote VICE on this one because I'm a complete tosser, but it's such a legendarily awful place. You can feel your soul dribble out your toenails as someone tries to sell you a bong with Che Guevara's face carved into it for the ninth time in half an hour.
Every city in Europe has a market area where crusties sell John Lennon posters and i-Pod T-shirts next to rudeboys hawking fake hash, but this is the only one so unpleasant that it’s tried to burn itself down twice in half a decade. Camden Town is the place where scenes goes to die. It is a machine designed to prove parents right about youth culture – it’s tacky, cheap, commercial and self-important.
White Cube
Bermondsey, SE1
I've always found White Cube surreal for some reason. It's a pretty normal art gallery, quite similar to MoMA's PS1, but it is like walking into a white-walled future in which everyone walks around pondering the futility of human existence while looking at some of Andreas Gursky's less good works. You should probably go to the Tate instead. Also Bermondsey as an area is super-chill while perhaps not being too interesting.
Most importantly though, just use your feet. The best part about London is it's extraordinarily nucleated and you can get anywhere with 30 minutes, a graffitied £20 with a Hitler-queen hybrid on the front and a couple of stale Hobnobs in your back pocket. Don't rush.
If you do need to rush, get a bike. They're pretty fun too, but don't get yourself killed because then the city comes to a standstill.
If you're in even more of a rush, grab an Oyster card (£5 deposit, £5 minimum top-up, completely worth it if you're going place to place a lot), and descend into the interminable furnace that is our public transport system, which was designed so that every year the temperature rises by 5 degrees each summer so as to solve our rampant overpopulation problem. Paying £8.60 for a Travelcard is silly, even for a short trip. Take it back to the ticket office when you're done and they'll give you your fiver back, or, alternatively, fall in love with the city and pledge to never lose it for when you inevitably crawl back.
Most importantly, have fun and avoid Camden. It's an amazing city and I wouldn't want to be anywhere else.