First Time London traveller

London is not a cheap city but you can save money. What does "very cheap" for you mean? As you admit you haven't a clue about what's close to what I'd suggest using Google Maps satellite view and typing in Big Ben London, United Kingdom. You'll be able to zoom in and out as well as hit search nearby. If you look closely you'll see that Big Ben, Parliament, Westminster Abbey and the London Eye are all near one another. Short stroll up Birdcage Walk past St. James Park and you'll be at Buckingham Palace. If you are at Parliament Square and walk up Whitehall you'll pass the Horse Guards, #10 Downing Street enroute to Trafalgar Square where the National Gallery and St. Martin's of the Field. A few more blocks past Trafalgar is Leicester Square. From there a few more blocks and you'll be in Covenant Garden. VERY walkable city with many of the "must see" sites within close proximity. Would be a tube ride to the Tower of London. FYI, I can't imagine these spots haven't all been reserved but check anyway. http://www.lse.ac.uk/lifeAtLSE/accommodation/home.aspx People may cancel. Otherwise see if you can get a room in someone's home through Airbnb. Make certain you are in travel zones 1 & 2 on the London Underground Map. http://www.tfl.gov.uk/cdn/static/cms/documents/large-print-tube-map.pdf That way you aren't paying more to travel into the city. As far as neighborhoods this link provides description. http://www.tripadvisor.com/Travel-g186338-s204/London:United-Kingdom:Neighborhoods.html I like Bayswater for it's diversity, cost value and the convenience of 3 tube stations to choose from. it's also close to Paddington Station which is where I arrive having spending a small fortune on the Heathrow Express. My choice. I don't dawdle when I travel. Want to get into town in 15 minutes, not interested in saving the money taking the hour tube ride but that's just me. You can get fresh sandwiches, fruit, soups and salads at Pret a Manager's. Also at Marks and Spencer's, Tedsco's or check out having some pub grub. Sign boards along the sidewalks and pathways, tell you the specials of the house. Loads of wonderful ethic restaurants to choose from as well. Purchasing and reading Rick's London Guide book would be invaluable to you.

Most of the usual tourists haunts are within Zones 1 and 2, so check that Tube map at the TFL site, as Claudia suggested.

London is very expensive. Don't expect to find any real bargains, especially in summer. In adddition to B&B's, and basic cheap hotels, a range of economy hotel chains have properties in London. E.g., Premier Inns. I've stayed at Premier's a couple ot times and found them satisfactory, but not fancy. Googling "cheap hotels London" should turn up more than you probably want. (I like to book directly with a hotel/B&B using their site or just call them.)

(The Bloomsbury neighborhood -- where the British Musuem is located -- contains a fair number of basic inexpensive hotels: Walk-ups, no restaurants but probably some kind of breakfast, etc. Bloomsbury is a convenient location with shops, cafes, etc.)

August is only 6 weeks away. Book as soon as you can.

Question for Forum Contributors: Is it still correct that, in general, two of the most expensive cities in Europe for lodging are London and Venice?

Check your dates with the Lancaster Hall. Fabulous location, good value.
http://www.lancaster-hall-hotel.co.uk/

For eating I recommend Pret a Manger if you're on a budget. They have refrigerated cases of all kinds of pre-packaged foods, mainly sandwiches and salads, and lots of other things. The quality is good, the prices are fair, and they are everywhere. Better yet you save time by not sitting down for a meal for too long! More time to sightsee!
Lodging is just going to be expensive unless you stay with someone...airbnb has lots of options where you can stay in a room in someone's home. I've done this quite a few times and I really love it! It's usually cheaper than a hotel or sometimes even a hostel, and you have the added advantage of staying with someone who knows the city and can give great advice on what to see and where to go.airbnb website
If you stay in central London the neighborhood will generally be good - consider Notting Hill, Holland Park, Chelsea, Kensington, Battersea, Belgravia, Mayfair, Pimlico - those are all fairly central to everything although they aren't the cheapest areas. The problem is the farther out you go the more you'll have to pay for the tube anyway and you'll waste time getting into town.

First, you should get a guidebook, which will not only have hotel recommendations, but also listings of sights. While London is not Rick's favorite city, and it shows in his book to some extent, his practical and logistics tips can't be beat for a first time visitor, so I'd still get his London or Pocket London book (or his England or Great Britain books instead, if you're also going elsewhere in the UK besides London)

A website I'm fond of is EuroCheapo. Here's their London page: http://www.eurocheapo.com/london/. And here's their take on various London neighborhoods: http://www.eurocheapo.com/london/hoods/

Check out the YHA hostel near St. Paul's Cathedral, for you first visit:

http://www.yha.org.uk/hostel/london-st-pauls

Great location, 35 pounds for a private room. This neighborhood is right in the midst of sights you'll want to see: 1. St. Paul's Cathedral 2. Tower Of London, to name two. Easy to get around on the tube (underground). You can get to other sights using the tube. You will want to see 3. Westminster Abbey(full tour), 4.Big Ben and Parliament Buildings (a walk-by will do), and 5.Buckingham Palace (a walk-by or drive-by will do), 6. The British Museum (enough cool stuff on exhibit to keep your eyes popping for three full days). That's just a start. There's a lot to see in London!

The best way to get a glance at all these and more, for someone who has a short period of time in London, is the Hop On Hop Off Bus. It's a bus you can catch at any tourist attraction. You pay, get on; if you stay on the bus for two hours, you'll get the whole tour around London, with a guide telling about the sights you're passing. Your ticket is good for 24 hours from time of purchase. So you can get off the bus anytime you want, enter a sight you want to see, come back out, get back on the bus, continue on until you see another sight you wish to explore.

This is a good way for a first time London visitor to get a quick overview of all the sights, and decide which ones look interesting enough to "Hop Off Hop Back On" the bus. It also comes with a free river cruise past some of the landmarks on the river. http://www.theoriginaltour.com/

Also, just walking is a great way to see some neighborhoods. London Walks is a company that offers guided walks for $7 to $12; www.walks.com.

For your second visit, I would recommend Premier Inn London County Hall. It's near the London Eye and on the River Thames basically across from Parliament and Big Ben. Can't be beat for location, price, and should be safe, but there are never any guarantees of that.

http://www.premierinn.com/en/hotel/LONCOU/london-county-hall?cmp=GLBC

If that hotel is booked up, try another of their locations, also a good choice:

http://www.premierinn.com/en/hotel/SOUANC/london-southwark-borough-market?cmp=GLBC

I second the advice given by Harold, that you should get a guidebook. Rick Steves Pocket London seems perfect for your trips. It can be ordered here on this website.

If you can afford it, use a Rick Steves consultant. The one I talked was very helpful and very knowledgeable. It is $75 for a half-hour over the phone and was worth it for me.

If money is a concern neighborhood supermarkets like Sainsbury Local, Tesco Local, etc., as well as Boots the Chemist and Superdrug drug stores have "lunch deals" of a sandwich, dessert or choc bar or crisps, and a drink bottle for £3.49, £3.99, etc., which will likely work out less than a comparable (but most excellent) Pret a Manger.

It costs more for hot food to go, and all food to eat in rather than take away because VAT is added in for what the government has deemed is restaurant eating. Therefor before you pay you will be asked if you are eating in or take away. Answer honestly.

Trivia: Pret a Manger was previously partly owned by that Golden Arches company but the marriage was not made in heaven.

Other Trivia: Pret a Manger make everything (well nearly) fresh every day and what they don't sell of the fresh food is given to food banks and homeless every day.

Still more Trivia: Pret a Manger regularly earmark one or more of their sandwiches as a fundraiser giving so much per item sold to a particular charity.

About a month ago we stayed 6 nights at Morgan House in Belgravia in Victoria Station area. It was very clean and safe and friendly. It was a bit small for us, but they mostly had rooms with shared Bathrooms for very cheap. We had one of the biggest rooms with bath and it was 108 a night with breakfast. But the shared bathrooms were much cheaper. Very great location just down the street from the underground. We did well with food, eating at Pret and Manger and Eats and there is a great pub a block away from Morgan House. When we got home we were surprised how well we did. I was a bit worried during the trip because everything I read said how expensive everything in London was, but we actually did much better than I thought including a couple of very nice dinners.

Can't help with where to stay since we rented a flat in London but my husband & I ate our meals at whatever pub was handy for where we were at meal time. The food was always very good and we ordered one meal plus two drinks and split the meal. We had plenty to eat that way. I'm not sure if that will be an option for you 2nd time but it did save us some money when we recently spent a week in London. Have fun!

I hope you'll also have a look at Rick's London guidebook(s) for a variety of trip details. Your local library may have a copy.