First off, London is not the money sucking city everyone has made it out to be. Sure, nice hotels and private transportation costs are at a premium, but if you are willing to ride the Tube, stay in an Airbnb, and eat some street food, you can experience all you need without lowering your credit score in the process.
Hop on Hop off Tour Buses
Yes they are cheasy and touristy, but if you spend your first day hopping on and off of one, you can see the majority of sites for the $33 ticket price, all without having to navigate the complicated Tube maps to get around. Pick the main sites track and get off at the sites that are most interesting to you (or all of them if you have time!) then when you are ready to move on, another bus comes within 15 minutes. The ticket also comes with a free river cruise so be sure to take that in the morning when it is less crowded and the lines are shorter. That is a lovely way to see the city from the water!
Recommendation: We use the Original Tour and it was pretty good, although their may be better ones out there, I think the major of them are the same and this company was the cheapest!
Tip: The buses can be crowded and you want a seat up on the top deck on the open side for the best view. You may have to ride to one stop while in a seat you don't prefer, then move to the better seats once a few people get off the bus. Also, if you buy a family pack of tickets you can save around 10 pounds.
Pubs - Surprisingly cheap!
Pubs, cheap? Yes! Compared to the average bar in the US, you can get much more beer at an English pub for less cash. Pints (which are more than our pours here in the US) run around 4-5 pounds and many pubs are also microbreweries so they make their own. The food at many historic pubs are also reasonably priced and you can get your traditional fish and chips and bangers and mash or shepherds pie if you are hankering for it!
Recommendation: Ye Old Cocke on Fleet Street. This pub is not only famous for it's long list of literary visitors, it's also the narrowest pub in London! It has modernized a bit but still has that mahogany bar feel and great beer and prices.
Accomodations: Go for Airbnb
If you are on a budget, accommodations will eat away at it fast. There are several entire apartments on Airbnb in nicer parts of town for rent. They do fill up fast though so be sure to start looking as soon as you book your flight! I like to book a listing that has some reviews I can read.
Recommendation: Look for a place near Paddington station if you are planning on going out of the city or to the further corners. It is a major train and Tube stop so you will have less connections when traveling, saving you time. It's also near one of the Hop On Hop Off bus stops!
Street Food and Take Away Restaurants
Who doesn't love a little street food? There isn't as much in London as there are in other cities but there are some, mostly snacks like nuts, ice cream, crepes, and other specialties. There are however many semi-street food stalls that are small Take Away Restaurants that have windows where you can get some quick fish and chips or sandwiches on the go. This is a great thing to do when visiting a nice square or park. Get something to eat and sit on a bench our on the grass and enjoy the atmosphere instead of being inside.
Recommendation: Get the caramelized peanuts. Trust me!
Get one of these babies if you are planning on using the Tube at all (and you should, taxis will rack up fast). You can add money on them and you get discounted fares. It also saves you time from having to purchase a ticket each time you use the Tube. You just flash this little card at the entry and again when you leave (so they know how far you've gone on the Tube) and refill as needed at the kiosks. It's also handy if you get on the wrong Tube line, you don't have to figure out which ticket you need to get back, just get back on the Tube!