Place I usually recommend to people visiting London

1. The Southbank Centre - I love this place. Besides libraries and museums, this superb cultural venue is another of those magical places where you're warmly welcome and where you don't need to pay to exist. Go to the 4th floor for a stunning view of the Thames from the balcony.
2. The Barbican Centre - the same, plus weird architecture and suspended walkways. Ask at the information desk about the passages you shouldn't miss.
3. Cyber Dog in Camden Market - just a really weird shop.
4. Little Georgia restaurant with Georgian / Balkan cuisine in Angel and Broadway Market. Ask for the ajika and plum sauces, mmm.
5. Dans le Noir restaurant where you eat in the dark. Expensive, but a truly memorable experience.
6. Paramount Bar. Best view I know of Central London from above. 360° of classiness. Go at night, it's magical. And you just need to get one drink, so it's cheap - but don't eat in the restaurant unless you're relaxed about a £40-50 dinner. (Phone booking needed - make sure you specify it's for the bar only -, plus relatively smart attire. But they're not that strict, don't worry.)
7. Hampstead Heath, Kenwood House and Brew House Cafe. Get lost in the woods, swim in the lakes, enjoy the wildlife. Gorgeous.
8. Indian Veg in Angel. Cheap Indian vegetarian all-you-can-eat buffet.
9. The Gate vegetarian restaurant in Angel and Hammersmith. Not as cheap, but worth it.
10. Mr. Falafel, the best falafel wrap in the known universe! Just go, you won't regret it, to say the least.