Bruntsfield and Morningside have the best selection of independent retailers away from the city centre.
The farmers' market on Castle Terrace each Saturday is a great place to pick up some high quality produce from some of the best local suppliers. I would also agree that Bruntsfield is a real gem, good tip.
We tend to end up shopping in Ocean Terminal in Leith, although closer to the Shore area are plenty of quaint and unique pubs and restaurants that make up for the high street chains at Ocean Terminal.
I have a homemade, illustrated handy guide to Edinburgh for sale at my website http://www.clairemurray.co.uk. Its full of great insider info, as well as a colour map. Check it out!