Now, the first thing you need to know about London is how to get there! Airports are the most common point of entry for most people and the trains are the next most used arrival points. Almost all my airport experience is through Heathrow which is the closer of the two big airports, the other being Gatwick (in addition to London City and Stanstead, but I have absolutely no idea about them.)
Arrival through Gatwick is probably less stressful. It's a big airport but not monstrous like Heathrow is. It's more modern and open. They have rail links between the terminals and the two main ways to get into central London are the National Express bus and the train. Both will bring you to Victoria train station, which is in the southwest part of London north of the Thames river. It's a large transportation hub with links to the city busses and the tube (i.e. underground or "subway" for us North Americans. Oh, and don't call it the Subway when you're there. They use that term for underground pedestrian tunnels!)
Gatwick Express trains to Victoria Station take about a half hour and are non-stop. There are trains to several other London train stations via First Connect that take a little under an hour for about half the cost of the Express. They serve London Bridge, City Thameslink, Blackfriars, Farringdon and St Pancras International, any of which may be more convenient to your hotel. Regular rail services into Victoria may also be cheaper, though slower than the Express. National Express busses can take an hour or more depending on traffic to Victoria Station and Easybus goes to West Brampton and Earl's Court.
Heathrow airport is massive with five terminals. There are shuttles between the terminals which you may need to get to the bus, train and underground stations. The cheapest way into the city is the tube or underground but if you have a lot of luggage, remember that you will have to carry it up stairs in the tube stations most likely as many of them do not have lifts or escalators or have a combination. It takes about an hour to get to King's Cross station and the tube, the Piccadilly line.
The most expensive mode of transportation is the Heathrow Express. It'll cost you about 15 pounds and it will get you to Paddington Station in 15 minutes. However, there is a new rail service to Paddington that takes about 30 minutes, making a handful of stops for half the cost of the Express. It's the Heathrow Connect and well worth looking into. From Paddington, you can catch the tube or take a bus or taxi to your hotel. National Express bus will take you from the central bus station at Heathrow to Victoria Bus station, also a good option. It costs about 5 pounds one way, which is nearly as cheap as the tube but doesn't get to the other side of the city.
Arrival at one of the main train stations will always put you at a tube station as well. There will be taxis and busses available outside the station and there will be machines and ticket booths for you to get public transportation tickets and passes if your hotel is too far from the station to walk.