As suspected, finding the “right” place to eat in London is a nearly impossible proposition, especially if you are a first-timer without a friend in the know. When in an international mega-city, it may just seem easier to find a sandwich place or a nice sit-down restaurant somewhere near where the rest of the tourists are flocking. If you don't value authenticity or care about budgeting your money, then you can stop reading now and find something more relevant to read. For those of you still around, there is a massive array of good, inexpensive food to be found from an international gamut of cultures, but you'll have to get away from Piccadilly Circus and the chains that surround landmarks like the British Museum. If you've got the time, then get out of central London and explore suburbs like Southwark-Lewisham and Lambeth (particularly the Brixton Market) for cheaper, comparable alternatives.
As far as genuine English food, you'll find traditional pub dishes like fish and chips all over the place. If that's all you want, close your eyes, turn in circles, and then open them. There will probably be a place that serves the dish in your sight lines. Pie and mash (a cold meat pie and mashed potatoes) is an age-old tradition; there are plenty of eateries that specialize in this blue-collar staple, although M Manze is the oldest and has a handful of central locations, while Clark's in the Exmouth Market is a must according to the locals. Other places may sell this dish (and the accompanying “liquor” gravy), but they are not authentic shops.
There are some excellent neighborhoods where the food is good, varied, and the tourists don't outnumber the locals. As you've guessed, they aren't too close to the landmarks that it is your duty as a traveler to visit. Still, Charlotte Street and its excellent range of quality cuisine is just north of city center, while Brick Lane in the East End is great for Indian food and other inexpensive dishes. Then again, Indian food is trendy in London and therefore can be found almost anywhere, but the Wembley neighborhood has a particularly high concentration. Another street worth mentioning is Upper Street, also just north of city center in Islington. There's a great range of food here, and a number of cheap and moderately priced choices.
For further information, especially since this write-up doesn't do true justice to the number of good eateries, cafes, and pubs that are thriving in modern London, peruse our activity-specific pages and start putting together an itinerary that excites you.