I've been all over Scotland for the golf, and here's my thoughts. Feel free to contact me privately if you want more.
As a single, you won't have any problem getting on the Old Course, simply show up early and be prepared to wait. If you have a twosome or foursome, you have to win the lottery - the procedure is on the Links Trust Website. Make sure you take the time in the Auld City to go to the Cathedral ruins and visit Young Tom Morris' grave.
Then get out, and go play other courses. My favorites are North Berwick West, Prestwick, Machrihanish, Crail Balcomie, Nairn and Royal Dornoch. Obviously, how many you can play depends on how long you're going to be there.
Before you go, take the time to study golf history! One of my favorite experiences was playing Musselburgh Old and stopping in Mrs. Forsman's pub for a pint, just as Young Tom and Old Tom did when they were playing a challenge match against the Parks. Or finding the Old Golf Tavern in Bruntsfield.
Far and Sure!