Oh! I'm so envious. I miss Edinburgh so much!
The Edinburgh bus tours are a really great deal. You can buy a ticket and jump on and off at any stop you'd like and they run regularly throughout the day. Be sure to check out Edinburgh Castle and the Royal Mile. If you're up for a nice walk, Arthur's Seat has some amazing views of the city. Take a day trip to Glasgow or St. Andrews. There are some really great day tours you can find through the visitor/information center on Princes street. Take another day to explore and shop along Princes Street. Duck into a number of museums if you get tired from shopping. One of my favorite pubs was Jekyll & Hyde's. See if you can find the bathrooms sober so you know where to look for them when drunk.
Find a rugby club's pitch and watch a practice or visit Murrayfield Stadium. Say hello to my old flat in Morrisson Circus near Haymarket. Best way to meet people and possible travel companions is find your nearest pub (not hard since they're everywhere!) and make some friendly conversation with folks. Oh! And take full advantage of the late-night chip shops!
Enjoy your trip for me!!!