Pack layers, vest tops with t-shirts etc as a lot of the shops etc get really warm. That way you can shed layers!
Edinburgh is such a cool city, go down to the grass market for a few drinks in the pubs there, go for walk up to Carlton Hill at sunrise or sunset (don't go alone though, as in any city ) if you're fit go up Arthur's seat and the view is a-mazing.
There are some great shops, theatres (the Bedlam usually has cheap improvised student comedy) if you have time and access to a car or the train, go to North Berwick which is pretty.
Biggest tip of all, wear comfy boots or shoes as Edinburgh is hilly and busy. Take time out to see some of the art galleries and take the weight off your feet in the coffee shops.
Lived there as a student for 6 years