Fife is a peninsula between the Forth and the Tay, and it has always had a slightly self-contained quality — its own identity, its own dialect, historically its own parliament. It's within easy reach of both Edinburgh and Dundee but doesn't particularly feel like a suburb of either.
St Andrews is the main draw and earns its reputation. It's a small town but manages to be simultaneously a world-famous golf destination, Scotland's oldest university town and a proper seaside place with good beaches. The Old Course is the most famous golf course in the world and is a public course — you can book a tee time through the ballot system, though demand is significant. The Swilcan Bridge on the 18th, the Himalayas putting course next to the first tee, and the general layout of the links around the town all make sense once you're there in a way that photographs don't quite capture. East Sands and West Sands are both good beaches; West Sands is the longer of the two and was used in the opening sequence of Chariots of Fire.
The cathedral ruins are worth an hour — it was once the largest church in Scotland and the scale of what's left gives some impression of what was lost. The castle, on the cliff edge, has a medieval mine and countermine system you can crawl through, which tends to be remembered more than the rest of it.
The East Neuk coast, running from Elie to Crail, is a string of fishing villages with red pantiled roofs and harbours of varying size. Pittenweem is still a working harbour. Anstruther has the Scottish Fisheries Museum and what many people consider the best fish and chips in Scotland. Crail at the tip is the most photographed. They're all worth a slow drive or, better, a walk between them along the coastal path.
Dunfermline was Scotland's capital for a period and has the abbey where Robert the Bruce is buried. It's a workaday town but the abbey and the associated ruins are genuinely significant. Culross, nearby, is a preserved 17th century burgh maintained by the National Trust — an entire streetscape of the period, which is unusual enough to be worth the detour. North Queensferry, under the rail bridge, gives the best views of the three Forth crossings.
Kincardine, Rosyth, Leven and Cupar are functional Fife towns without huge visitor attractions, but the county as a whole has enough variety to fill a few days comfortably.