It is a good idea to have an idea of the sites you want to visit, as well as things you want to do during your trip. Do some research on travel blogs and communities. You should ask friends and family who have traveled before about places they enjoyed the most. Read through site-specific guide books like Lonely Planet, Let's Go, Rough Guide, Rick Steves, or Frommers.
Although you will want to book all travel as far in advance as possible, you should NOT book any trips until after the first week of classes. You may find that there are already NYU-sponsored trips in place.
If you are planning weekend travel, then flying from city to city may be your cheapest option. With flights, the earlier you book, the cheaper it will be. Be aware that the taxes for the flight may be more money than the cost of the actual flight. Many of the cheapest flights leave very early in the morning from airports outside of the city.
In some places it can be difficult and expensive to travel to these airports at these very early times because public transportation may not run all night. If you go on a low-fare airline, expect to pay a fee for any stowed luggage you bring.
This site is specifically for train travel within the United Kingdom
This train will take you into the heart of Europe.
Railpass lists all of your train options and information about the trains for all the different European countries.
If you are planning on traveling (especially after your semester) you may find that Eurail is the best option. It is a train pass that gives you train travel within 18 European Countries. Not only is it a cheap, convenient way to travel around Europe, but also many Euro trains are super comfy! Buy your tickets in the US as they are more expensive in Europe.
This site is specifically for bus travel within London, United Kingdom. Here you can plan trips and see bus routes and maps.
With Megabus, you can travel throughout the UK for as little as £1.50.
Busabout is designed for back-packers which makes it a great way to meet people, and a cheap way to get around—it may be more appropriate if you plan to do more traveling during the summer.