I'd definitely take a laptop for a 2 week trip as it makes planning and locating touristy things that much easier. Try and stay in a place with free wifi - hostels will usually have free wifi, hotels usually charge. Free wifi options: http://londonist.com/2007/05/free_wifi_in_lo.php

For getting from Heathrow to London - if you're staying near Paddington I'd suggest the Heathrow Connect instead of the expensive Heathrow Express. They both run on the same track but the Connect is less frequent (every 30m and has a couple of stops so takes 10 mins longer) but is almost half the price. Alternatively take the tube (subway) if you've got the time to spare or aren't staying near Paddington.

Transport: the tube is usually the easiest option, though often it's quicker walking if only one stop. If staying out, tubes stop around 00:30 (earlier on Sunday nights) but there are nightbuses - see http://journeyplanner.tfl.gov.uk/ for planning a bus route to/from your hotel. For tubes/buses: you can buy a (pay as you go) Oyster card when you get here (£3 refundable deposit).
Cabs: you can only hail black cabs. Other cabs (usually cheaper) are called "minicabs" and can only be booked by phone. Addison Lee is the biggest minicab company but not the cheapest.

http://www.ianvisits.co.uk/calendar/events/ is good for London geeky events - worth checking out when you're here to see if anything interests you. Other people have covered the museums so I won't bother.

Worth a read: http://wikitravel.org/en/London

Buy a copy (in London) of Time Out London (weekly events/gigs/film/tv/theatre etc listing magazine) and also one of their London guidebooks. They're the best guides. May also be worth picking up a free "TNT" magazine (they're everywhere) for gig/party listings - is popular with backpackers/younger people.

Try some markets in London too: Borough Food Market on a Sat morning; perhaps Camden Market ("alternative" but rather tired) on Sat afternoon. Spitalfields on a Sunday (trendy junk). Columbia Road flower market on Sunday morning is very good too (for something a bit different). There's Christmas markets in Hyde Park and by the O2 (North Greenwich tube).
Main deparment stores are between Oxford St tube & Marble arch tubes - just walk along Oxford St. Selfridges is more popular with Londoners than Harrods. If you want a shopping mall then there's Westfield (Google for locations).

Get a single ticket on a boat to Greenwich to see the meridian line, naval observatory, painted hall etc. But don't take a boat back as it takes forever - instead get the tube back.

Out of town: Brighton is also worth a weekend trip (train is 1hr from Victoria or London Bridge). Sat/Sun are best for the shops on the lanes. It's the closest seaside town to London - though take a map so you know how to find the lanes (on left hand side as you walk from station to seaside). December won't be so great but it's still worth a visit.

You can go to Oxford, Brighton etc on the bus but I prefer the train (quicker, more comfortable). I'd recommend pre-booking at least a day in advance - when you book you can choose to pickup the ticket from a machine at the station. http://www.eastcoast.co.uk/ is a good site for getting an idea of rail fares - if they're too expensive then take a bus (most buses leave from the coach station near Victoria station).

I spent 3 weeks backpacking around Iran in May last year.

Their blocking system is fairly limited. Each ISP implements its own set of manually updated filters (not a central blocking system like China). I was trying to access certain sites -- www.sitename.com might work at one place but be blocked at another, though at the other net cafe sitename.com or IP address would often work just fine. I found the blocking policy inconsistent, though not that many sites were blocked (mainly gay sites were blocked).

Because of the Iran/Iraq war the population is very young -- 70% of Iranians are under 30 according to the Iran Lonely Planet guide. I imagine that'll mean plenty of blogs, whether insightful or the usual blog trash. People were quite politically aware and well educated. The news media seemed no more biased than Fox News in the US!

It's a beautiful country and well worth a visit. Persepolis is amazing. Tehran was like any other big city -- lots of expensive houses, cars and more liberally dressed women. The latest model mobile phones were available everywhere. I was offered alcohol quite a bit (especially by taxi drivers). It's illegal for Muslims to drink but the Christian and Jewish population are able to drink. Incidently, Iran has the highest Jewish population in the Middle East outside of Israel.

I didn't know much about Iran before visiting, I'm just glad I went. Unfortunately if you're American/British it's difficult (though not impossible) to get a tourist visa unless you're in a tour group. I presume this is due to reciprocal restrictions applied by the UK/US on their citizens.

The last few years there have been fairly low in terms of tourism numbers and people were incredibly friendly to me - offering to take me to their homes for dinner and so on. Plenty of people were critical of their government but were just as critical of the American govt.

Funnily enough I just visited Israel last month and had a 45 minute interrogation because I'd visited Malaysia (a very westernised 70% muslim country). I'm glad I wasn't using the passport with the Iranian stamp in it!

I took photos of the nuclear installation between Kashan & Abyaneh despite the taxi driver panicking I'd get caught (you're not allowed to take photos of military installations). Though you can get a much better view of the place through Google maps!