The fastest way to get to Edinburgh from the United States is to fly into Edinburgh Airport (EDI). There are direct flights from major US cities like Atlanta, Orlando, and New York.

If you prefer not to fly directly into Edinburgh Airport, you could consider flying into other nearby airports such as Glasgow Airport (GLA) or Aberdeen Airport (ABZ) and then taking a train, bus, or rental car to Edinburgh. Glasgow is about a 1-hour drive or a 1.5-hour train ride from Edinburgh, while Aberdeen is about a 2.5-hour drive or a 2.5-hour train ride.

You can also opt to fly into London, which often offers less expensive fares. If you're flying to Edinburgh via London, you can land at any of London's major airports: Heathrow (LHR), Gatwick (LGW), Stansted (STN), Luton (LTN), or London City (LCY). Once you arrive in London, you have several options to travel to Edinburgh:

1. **Train**: You can take a train from London King's Cross station to Edinburgh Waverley station. The journey takes around 4.5 hours, give or take a few minutes depending on the service.

2. **Bus**: Several bus companies operate services between London and Edinburgh, with journey times varying from around 8 to 10 hours.

3. **Car**: You can rent a car and drive to Edinburgh, which takes around 7 to 8 hours depending on traffic and the route you take.

4. **Domestic Flight**: Alternatively, you can catch a domestic flight from one of London's airports to Edinburgh, with several airlines offering frequent services between the two cities.