Edinburgh is a stunning place!
I truly fell in love with this city. I was suppose to only spend 4 nights but could not get back on the train to leave. So ended up spending 7 nights. So much history, culture and activities. You'll never have a dull moment. Evrything is in close proximity so there's really no need to use the public transportation unless walking a little is a problem. The train station is in the heart of the city which makes exploring some smaller towns outside of Edinburgh very easy to do.
Insider Tip:View the underground city!!! Such a cool experience and history lesson. Arthura seat provides amazing views but is a long and tiresome walk. Can take a few hours to do depending what trails you take. But can be done in an hour (maybe little longer) if you go strait up and down. Stirling is a small town 45mins away with a great castle and St. Andrews is a 2 hr ride from the city. If possible take the bus to St. Andrews. The views are much better then train travel and roughly same travel time