As a local I don't really visit the tourist attractions (or at least haven't visited many for years) so I'll leave them for others to recommend.
For getting around, an Oyster card is essential. You don't need (or want) a car, and you will pay way over the odds for your transport if you don't have an oyster card - you pay a £5 deposit for the card, which you can get back by returning it once you are finished with it. Lots of Americans seem to have something against public transport (apologies if you aren't American!), but at least in London it is efficient, safe, and 24h. Everyone uses it, and it is generally the cheapest and fastest way of getting around the city centre.
Weather will be cold and rainy but mostly above freezing. Snow is unlikely - and if it does fall, it won't last.
For upmarket shopping head to Mayfair and new Bond Street (Green Park tube). For major international chains and lots of flagship stores, Oxford Street and Regent Street (Oxford Circus tube). Harrods and Fortnum & Mason are also good for luxury items.
As far as restaurants are concerned, there are many excellent ones - far too many to name, so details on where you are staying and what you are into would be useful. Restaurants nearest to tourist attractions are usually overpriced and disappointing; if you go a few streets back you will likely find places that are cheaper and better.
Crime is minimal, and if you keep your wits about you (i.e. don't flash cash about, don't put valuables in your back pockets in busy areas etc.) it won't be a problem. I have lived here all my life, and I have never been pickpocketed or had anything stolen. Once someone tried to mug me, but I just beat him up.
If you have an iPad and want a data plan, all 5 major phone providers will give you a free sim with a one-month no-contract package. You can get 1GB from £7.50, and 10GB for around £25. 3G is fast and almost universally present, and access to maps is useful since if you are used to cities in grids, London will confuse the hell out of you.
Let us know where you are staying and what you are and aren't into, and you might get some more specific/useful tips. Plus, as a resident I'm happy to answer other specific questions you may have.
Oh, and don't get stabbed.
http://www.guardian.co.uk/uk/2011/dec/26/man-stabbed-death-oxford-street?intcmp=239