Having spent considerable time in Edinburgh over time and also having checked with someone who lives there at the moment the following are
recommended restaurants:
The Grain Store
30 Victoria Street, Old Town, Edinburgh, EH1 2JW
Oloroso - top floor modern restaurant on George Stree - great views of castle
33 Castle Street, New Town, Edinburgh, EH2 3DN
Stac Polly -there are 3 in edinburgh - the main ones are 29-33 Dublin Street, New Town, Edinburgh, EH3 6NL
or 8-10 Grindlay Street, West End, Edinburgh, EH3 9AS
The Witchery by the Castle - this is more expensive but very popular Castlehill, Royal Mile, Old Town, Edinburgh, EH1 2NF
David Bann Vegetarian Restaurant - excellent veg restaurant 56-58 St Mary's Street, Old Town, Edinburgh, EH1 1SX
Fishers in the City - Nice fish and general restaurant - very central 58 Thistle Street, New Town, Edinburgh, EH2 1EN
The best website to check is
http://www.list.co.uk/ which details whats on and what restaurants are in edinburgh
By the way there is a very effiicent bus service to the city centre from the airport if you don't feel like forking out on a taxi (although the taxi fares aren't that steep and if its your first visit to the city might be better to get one until you get your bearings)