I liked Mussel Inn in the New Town area of Edinburgh between Princes and Georges Street. Relaxed atmosphere, casual, good seafood at good prices. If you're looking for a casual restaurant. Get the half kilo of mussels with the creme and shallot sauce. Also, try to book reservations if you're planning on going during a weekend. When I went, we were fortunate enough to arrive early (6pm) for a seat but by the time we left it was full.
It's not a restaurant but I really enjoyed Oink on Victoria Street which is a takeaway/casual dining. If they're huge roast pig in the front window does NOT want to make you go in, then I don't know. ha Get the apple filling with the skin if you get the roast pork sandwich. It should cost you about £4.
I also enjoyed The King's Wark over in the Leith area of Edinburgh. Its neither in the Old or New Town area and you'll need to take a short 15 minute bus ride to this area. It has a cosy atmosphere with solid, good Scottish food. I went for a scottish breakfast but have heard their other meals are delish as well.
I hope these recs help.
Rose St, Edinburgh, City of Edinburgh EH2 2, GB
The King's Wark