If you travel to Edinburgh with your dog(s) and you don't go to Braid Hills then you have missed something most marvellous. It is a 15-minute bus ride from the city centre (11 15 16 or check with Lothian Buses). There are 2 golf courses which offer great heathland rambles and have magnificent city / castle / Arthur's Seat views; but across the road to the north [i.e. nearer city centre] (closer to Morningside bus stops) you will find multiple woodland paths circling a beautifully clean river.
If you own or manage The Braid Hills, register now for free tools to enhance your listing, attract new reviews, and respond to reviewers.