I'd recommend the Bow Bar, the Blue Blazer, The Brauhaus and The Cloisters. All of them have a good selection of craft beer from Scotland and elsewhere, also unlike Brew Dog they have cask beer too.
The Bow Bar is on Victoria St, basically that steep curvy street rising from the Grassmarket towards the Royal mile. The Blue Blazer is near "the triangle of sin", it's on Spittal St. which you will find if you are walking from the Grassmarket towards Lothian Rd. The Cloisters, which was my favourite bar in Edinburgh (BrewDog aside) is on Brougham St. and about five minutes walk from The Cloisters, just turn left up Lothian Rd, and it's just past Tollcross.
For bottled and imported craft beers The Brauhaus is the place to go, and it's just one block away from The Blue Blazer, on Lauriston Place.