The bus trip between Brighton and Edinburgh is not for the faint hearted as it takes between 12-13 hours to make the trip including a transfer at Victoria Coach Station, London. If pushed for time you may well wish to consider taking a plane from Gatwick Airport which is not far from Brighton. The National Express coach leaves from the Pool Valley Coach Station, which is off the seafront just west of Brighton's pier. National Express runs one coach a day plus one overnight service. A single ticket costs approximately 46 GBP or a return is 55-69 GBP.
After booking online, you will be issued with an e-ticket, which you need to print out and take with you, or an m-ticket, which is sent to your mobile phone. Otherwise you can get at ticket at the bus station.
If you are looking for the cheapest option you could consider getting to London by train or bus then taking the Megabus route from London to Edinburgh, as fares from London to Edinburgh can be as little as 5.50 GBP for a single if you book far enough in advance.
The train is twice as fast as the coach. A single ticket costs about the same on both, and you might be able to get a good deal on a return train ticket if you book well in advance of travel. Trains run twice an hour from Brighton Station to London's Victoria Station, from where you need to transfer to King's Cross for your onward connection. To get to King's Cross, take the northbound Victoria (light blue) underground line. Allow extra time for negotiating the Underground system. Subway trains run every 2-3 minutes but there are often delays. Try to time your journey so you avoid travelling across London in the rush hour - subway trains can be impossible to board, especially if you are travelling with luggage. The total journey is approximately 6 - 6 ½ hours. An off peak return ticket is 125.90 GBP, or a single is approximately 49 GBP.
The UK train ticketing system is notoriously complicated. It is usually cheaper to buy your tickets up to 6pm on the day before you travel than on the day. And fares are more expensive if you travel before 09:30. You get the best fares by booking up to three months in advance. You can buy your tickets online, at the station ticket counter, or from the ticket machines on the main concourse. If you book online you will need to collect your tickets from the ticket machine using the same credit card you used to book your tickets online.