There are several ways to travel between London and Edinburg. The train is probably the most relaxing way of doing the trip, since the stations are in the center of both cities and there are no big complications to carry normal luggage.
Flying is a possibility, there are several low cost carriers, but one has to add extra time to get to and from the airport, and to go through security and check-in.
The train is an excellent option, if one buys it in advance it is possible to get quite affordable rates. It takes 4 hours and a half from London. The train leaves Kings Circle and arrives in Edinburg to Waverley station.
The cheapest train tickets are released normally 12 weeks before the date of travel.
One can get train tickets directly with the rail company or in the USA and other countries it is possible to buy these tickets in several different specialized websites or with the help of a travel agent.
Another good advantage with train travel is that both train stations are well connected with all kind of public transportation options, which makes it easy to move around the city with a worry free experience.