I suggest to go to London for two weeks, visiting not only the central district but also suburban places like Hampstead, Richmond, Wimbledon, and Kew Gardens.
The "Big Red Bus" and "Original Bus" tours are a great way to see London and top sights. Some routes have a live guide and some have recorded conmmentary. Busses are double decker, some with open tops. Pay one price for the day, hop on or off at any stop, Thames river tour included in price.
London is incredible. You should spend at the very least a week there. All the museums are free and the locals are all friendly and willing to help. Getting around is very easy when you use the underground. Go to Brick Lane and have some drinks. Also, for the ladies, go to Canary Wharf around 5 p.m. on a Friday -- there are loads of men at the bars who just got out of work!
Do the Red Bus trip around London. I gave it a miss because of the cost but someone else I knew did it and loved every second. It seems like it would be a great induction to the city to help give you a feel for where everything is and makes it a good way to pop in and out of some of the tourist attractions that you don't need a lot of time for.
Heathrow is the main airport for London and is to the west if the Greater London area. There are various types of public transport varying in cost. The budget minded should take the underground (tube, metro, subway) but note there are fast trains which cost more. It's not a bad idea to have a night here to acclimatise to London. There are places that you might visit from here (do your own research!) and London buses currently charge a flat fee of £2.
Luton airports sometimes have cheaper flights and there is an amazing hotel there called the Red Lion Inn that is super cheap and about a five euro cab ride to the airport or the train station. We took the train from London to Luton and then flew out from there.
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