Visit the O2, eat at Zizzi's (have a bottle of the house red), go into any pub and ask for a Czech lager with a tub of chips covered in malt vinegar, grab a Starbucks in Greenwhich and walk along the Thames, or grab a Starbucks from London Bridge and walk along the Thames, get a travel card for the zones you need (1-4, I'd wager); public transportation will get you absolutely anywhere, DLR, underground, overground and trains are your best bet for longer distance travelling. If you want to see the city, walk rather than take the bus. There are maps on almost every street corner in the centre of London. If you do take the bus, say thank you to the driver when you step off.
Most of the crime that takes place is gang-related, and even if you frequently visit places like Camden, Brixton, Tottenham and Hackney, it's very rare. I live in South East London and it's perfectly fine.
Remember that cars drive on the wrong side of the road and that jaywalking isn't illegal (if you go to somewhere like Oxford or Piccadilly Circus, you'll see people navigating traffic).
All in all, it's a beautiful city with tons to do and tons to see. Spending the evening at the pub or a restaurant is a big part of the local culture and what people do to enjoy themselves. If you walk along the south side of the river near Tower Bridge, you'll find many beautiful pubs right by the water. The view is gorgeous at night. I'd recommend either Zizzi's or Pizza Express (don't let the name fool you, it's a nice place) around those parts.
Also, there's an enormous M&M's shop at Leicester Square.