St Andrews is a small, handsome Scottish town on the Fife coast, about an hour north of Edinburgh, and for the size of the place it punches remarkably above its weight. It is the home of golf — the game has been played on the Links here since at least 1457, and the Royal and Ancient Golf Club of St Andrews, founded in 1754, is still the guardian of the rules worldwide. It is also the home of Scotland's oldest university, founded in 1413 and famous in recent years as the place where Prince William met Kate Middleton. And it was, for several centuries, the ecclesiastical capital of Scotland, until the Reformation left its cathedral and castle in ruins.
The town is compact — you can walk from end to end in twenty minutes along the three old parallel streets (North Street, Market Street, South Street) — and everything worth seeing is within that footprint. The cathedral, begun in 1160 and once the largest in Scotland, is now a dramatic set of ruins surrounding the intact St Rule's Tower, which you can climb for the best view over the town and the North Sea. The castle, perched on a cliff above the sea, was the archbishop's fortified palace and the site of some of the more violent episodes of the Reformation — you can explore the dungeon and crawl through the mine-and-countermine tunnels dug during the 1546 siege. Outside the castle walls, cobblestones mark the spot where the Protestant reformer George Wishart was burnt at the stake in 1546; his patron John Knox watched and was radicalised.
For golfers, the Old Course is the obvious pilgrimage, though getting a tee time requires a ballot, a wallet, and some luck. The British Golf Museum next to the Royal and Ancient is a decent consolation prize, and you can walk the 18th fairway for free — it is public land, and the Old Course shares the West Sands beach with dog walkers and the opening scene of Chariots of Fire, which was filmed here.
Beyond golf, the real pleasure of St Andrews is the scale: three long sandy beaches, a harbour full of fishing boats, university students spilling out of pubs on Market Street in red gowns, ancient ruins, and the North Sea light. It is easily day-trippable from Edinburgh, but worth an overnight to see it after the tour buses leave.