When you arrive at Heathrow have a passport photo and ask for a one week travel card. They'll set it up there and then. Makes for a nice little personalised souvenir also.
If you're going to be staying around the middle then zones 1 & 2 are all you'll need. From Heathrow just ask for an extension tickets to cover your journey into zone 2 where your card kicks in. Sounds more complicated than it is.
Anytime I was there the travel card covered overhead rail as well as London Transport.
Also, ask for an outer and inner London bus map (free), saves you buying one. Bussing it IMO is better than tubes around Central London simply cuz you'll see more.
Zones .....
Anytime you need to go outside the zones (which if you're living in 1 or 2) shouldn't be often just buy an extension ticket. Make sure you do that BEFORE your journey or they can hit you with a fine for fare avoidance should you try and pay at the end.
If you're going with one day cards remember that they can't be used on weekdays before 09:30
To add to the other suggestions .......
Try speakers corner (Hyde Park Corner) on a Sunday am/early pm. It's basically a unique place where people go to rant. When I was there it was Arabs going off on Jews and vice versa although it might have toned down a bit with hate laws being introduced.
Camden Town on a Sunday for the market, pubs and people watching. It's a hive of activity although I'm not sure how it's recovered from a huge fire last year.
Go to the dogs. Fun night out for a group and something to do over a few beers. If a dog has a crap before it goes into the traps, then despite what others say, bet against it.
Few tracks scattered around London.
Tour bus is worth the price as 4 x 4 says (edit not) just for the scale of the city, the river tour down to Greenwich is worth doing.
Definitely buy Time Out magazine, it'll have all the listings for everything (sport, comedy, museums ...) at your fingertips.
Gunners/Hammers
should be a really good atmosphere.