I was just there this past September/October. The Underground will get you anywhere, right from the Airport...the Oyster Card is a must so that you can get anywhere worry-free. You can pick one up at the station at the airport.
I like 4X4's idea of doing the bus tour. There are a couple of Hop On Hop Off tours that you can take which will show you the hilights, and then you can figure out where you want to go in the most efficient manner from there.
Museums are cool and free, but most want a small donation (1 or 2 quid).
Take a day trip on the National Rail lines to Windsor or Oxford...or both.
re: looking right, most of the places that are busy have paint on the cross walk saying which way to look as well, which is nice.
Go to the shows & get tickets at one of the booths at Leister (sp) Square...but there are plenty of booths, so don't be afraid to shop around a bit. I'm assuming you're in your 20's/corporate, so I strongly recommend you go see "Avenue Q".
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