I notice lots of discussion on London restaurants, so as a Londoner, here are my suggestions. I have tried to include a range of areas, budgets and cuisines.
Lunches: Lunch menus are often good value at a fraction of the evening cost. Arbutus in Soho is great value and has one Michelin Star. Nobu on Park Lane (near Grosvenor House) is also a good pick. For fine dining, Alain Ducasse at the Dorchester is a relative bargain at around £55 for 3 courses including wine, water and coffee. This too is near Grosvenor House and Grosvenor Square.
Best food markets: I would recommend checking out Borough Market (near London Bridge). If you are staying at County Hall, it is a lovely walk along the river to the market (if it is not raining). On the way, you will also pass another market by Royal Festival Hall which holds festivals from time to time like the bread festival and chocolate festival. For a more cutting edge experience, slightly off the beaten track, head to either Broadway Market in Hackney or Brixton Village. Brixton Village is not strictly a market but a series of well priced restaurants in a former covered markets, they have good pizza (Franco Manca), Colombian, Mexican etc.)
What about British food: For the ultimate experience, don't miss Dinner by Heston Blumenthal but remember to book ahead. Beat fish and chips any day. Also, for a real British experience, heading to a curry house is a good idea (sounds strange but trust me). There are many good ones on Brick Lane (near the City) and Tayyabs is great value nearby (prepare to queue). For an authentic experience off the beaten track, head to Tooting (don't wander around the back streets too late though)
Steak: There are several branches of Hawksmoor and this offers the best steak in London in my view. Even better than the steak house at teh J.W. in Grosvenor House.
Breakfast: For a decadent breakfast, head to the Delaunay (Covent Graden) or Wolsey (Picadilly). Good value and one to remember. Alternatively, a greasy spoon is a very British experience but you won't find may authentic ones in town- instead head to areas like Bethnal Green.
Beer: For great beer, look for branches of the Craft and teh Cask (same company, different names). The one in Pimlico is great (near Victoria station). Alternatively, head to Kernel Brewery in Bermondsey on a Saturday afternoon (it closes around 4pm). This last one is a top tip, if you like beer, I recommend checking it out.
Casual but upscale: Sometimes it is nice to go somewhere with great food and atmosphere but without the starched table cloths. I would recommend Opera Tavern in Covent Garden or Bar Boulud (at Mandarin Oriental) in Knightsbridge.
Hope that helps and would be interested to hear your experiences.