There are three main ways of getting around Edinburgh: Bus, Taxi and Shanks's Pony (walking).
One of Margaret Thatcher's legacies to the UK was a totally fragmented, not to say fractured, bus service. Edinburgh is comparatively lucky, having only two competing services.
By far the biggest is Lothian Buses. As you will see from their route map they provide an excellent service into and out of the city. Routes 12, 22, 26, 30 & 31connect the city centre with Murrayfield Stadium. There is a £1 flat fare for adults for all journeys, 60p for children. You pay as you board the bus and no change is given.
If you want to do some sight-seeing whilst here there's the City Bus Tour by open-top bus, or you can consult the list of routes serving places of interest. If you plan to use the bus a lot during your stay you might consider getting a Day Ticket or CitySingle.
The other service is provided by First Bus. Since I always use Lothian Buses and find the First Bus website less than user-friendly I'll leave it to you to wrestle with it if you feel you must. Adult fares are 90p for short journeys and £1.10 for longer journeys. The city centre to Murrayfield is classed as a short journey. Once again pay on boarding and no change is given.
Taxis can be hailed on the street, or picked up at ranks around the city. Alternatively, you can call for a cab or pre-book from: