The city centre is best walked — almost everything you have come to see is within thirty minutes on foot from Princes Street. Just remember it is built on hills, and the Old Town is genuinely steep; comfortable shoes matter.
For longer hops, Lothian Buses runs an excellent network across the city; a single fare is a flat £2 (exact change only — drivers do not give change), or a day ticket is £5 and saves the bother. The tram runs in a single line from the airport via Murrayfield and Princes Street out to Newhaven, useful for the airport and Leith but not much else. Taxis are easy to flag in the centre; black cabs use the meter, private hires are usually a little cheaper. The city is reasonably bike-friendly, especially along the Water of Leith path and the disused railway lines that have been turned into cycle routes.