London on a Shoestring
As a native New Yorker I'd been a bit embarrassed that I've traveled so much but never been to London. I'd always heard it's more expensive than New York—and I was horrified at the age of 21 when I heard that they don't have the cheap food equivalent of pizza ("What do you eat if you only have $2??"). But I finally made it there and found that in fact, unlike New York, it's quite possible to do it cheaply.
I stayed in a hostel across the street from the British Museum called the Astor Museum Hostel. For 19£ per weekday night, you can get a clean bed in a shared room with large windows overlooking the museum. They have a daily breakfast of cereal and breads for 1£, the proceeds from which are donated to a local charity for the homeless.
The atmosphere is chill and respectful; this is not a party hostel. People are doing their own thing, but happy to do things together too. My first night I spent watching Netflix with at least 5 other people in the lounge, and it felt like chilling in a friend's living room. Another night we (a newly acquired "we") celebrated a young Russian's 22nd birthday—he'd bought a cake to share; we took him out for a beer.
London is easy for walking if you're staying in the central area (Kensington, Westminster, City of London, Southwark, Lambeth). Distances aren't too far, and the sidewalks aren't that crowded (again, I'm from New York so my scale might be a little different than other's).
The tube was efficient and reminded me of Paris's metro. But to me it was pricey for a single ride (4.90£ if you pay cash. Get an Oyster card and it's 2.90£ peak/2.40£ off-peak.). I preferred taking busses which is cheaper (2.60£ cash or 1.50£ with Oyster card) and are double-deckers, meaning they have an upstairs seating area. I made sure to ride up high every time that I could.
There are also hire bikes around the city which I'm sure work well and affordably, but I didn't try them (see why here).
Something I appreciate all around England are Tesco sandwiches. No joke. It was a revelation. Items that are coming close to expiry are put on a deep discount, so you can get sandwiches for 35p or even 15p, as well as sides or mains for 1 or 2£. Even if you don't find discounted items when you go, the regular price for an egg and cress sandwich is 1£—cheaper than a slice of pizza!
Sandwiches will do in a pinch, but if you're willing to cook, then hitting up one of the markets for produce is the healthier cheap option.
Museums. Unlike New York, museums in London are free, encouraging one to visit as many as possible. I swung on the swings that are temporarily installed in the Tate Modern and went to its observation deck for a view of the city. I saw the Rosetta Stone, the extremely well-preserved remains of a man who met with a tragic death, and the contents of a prehistoric treasure chest at the British Museum. In the Natural History Museum, I fulfilled a childhood dream of seeing dodo birds (dead, of course, sadly) and then continued to amuse my childish side by watching an animatronic T-Rex and gaping at the giant skeleton of a blue whale. Across the street at the Victoria and Albert Museum I admired clothing from the 18th-century to today and watched soloists perform arias for little children. I saw Elizabeth I and Henry VIII age over the years through their portraits hung in the National Portrait Gallery, and in the adjacent National Gallery, an exhibition of rarely-seen pieces by Degas.
Outside of the National Gallery is Trafalgar Square with fountains and a statue to honor Admiral Nelson for his victory in the Battle of Trafalgar. To the west is Buckingham Palace. To the south is 10 Downing Street, where the Prime Minister lives, and Westminster Abbey and Parliament. These all cost money (except Downing Street—police prevent you from getting close), so just take pics. Unfortunately, Big Ben is undergoing renovations so it's covered in scaffolding and will not resume its daily chimes until 2021. Big Ben sits next to Westminster Bridge from which you can take pics of the city by the water as well as of a newer addition to the historic skyline—the London Eye ferris wheel.