www.cambogardens.org.uk/ Cambo Gardens just a few miles out of St Andrews, near the Kingsbarns golf course, is beautiful. Lovely gardens, walks, and great vegetarian/vegan food.
Tentsmuir is lovely too. I prefer to park the Tayport side and walk along that way.
Falkland Palace, Kellie Castle, and a bit further afield, Culross are all worth visiting.
St Andrews Cathedral and climb St Rule's Tower.
St Andrews Aquarium - they've got a new baby seal there.
St Andrews Museum (in Kinburn Park) and the University museum on the Scores. Preservation Trust museum on North Street.
Jannettas for ice cream of course!
If you do go over to Dundee for the V&A, the McManus is (in my opinion) far more interesting! There's also an open top bus tour of Dundee and other places like the DIscovery ship, Verdant Works, Transport museum, waterfront area (and urban beach). If you like cycling you can hire bikes and ride along the riverfront - you can get along to Broughty Ferry which is lovely too.
Crail/Anstruther/Pittenweem are little coastal fishing villages - some nice places to eat and gorgeous views.