Many of the places you might visit are within walking distance. This includes most LSE and University of London halls. London is a very safe city, but obviously you should take precautions, especially if you are alone at night and or intoxicated.
The Tube, Overground, and DLR run from around 0530 until midnight, every day but several tube lines will run for24 hours on weekends from 12 September 2015. They cover the whole of London.
Buses cover even more of London and run all day and night across most routes.
Contactless payment is available on the Tube, Overground, DLR, and Buses. You can also use an Oyster card. If you use public transport every day, you may want to apply for a student Oyster card for a discounted weekly or monthly rate.
Licensed black cabs (taxis) are available to hail in central London at any time. Or you can use Uber or Hailo to ask for a driver to come to your location.
From Heathrow Airport, you can take the Heathrow Express, Tube (Piccadilly Line), or a cab or similar.
From Gatwick Airport, you can take the Gatwick Express or Thameslink services, before switching to the Tube or buses. You can also take a cab or similar.
For domestic travel, it is usually easier to travel by train/rail than air. You can plan journeys and buy tickets for all UK train operators from National Rail. The two major coach operators are National Express and Megabus.
For international travel, Skycanner is useful for comparing flight prices and times. Eurostar may be another option, by rail. Megabus offers coaches to some European destinations.