Québec City's main appeal is the Old City itself — walking the fortification walls, exploring the narrow streets of Vieux-Québec, and taking in the view from Dufferin Terrace. Most of what you'd want to see is within easy walking distance in the Upper and Lower Towns.
The Vieux-Québec walled city is the starting point for everything. Walk the fortification walls (about an hour, free), stroll the Dufferin Terrace for the panorama over the St. Lawrence, and take the Funiculaire down to the Quartier Petit-Champlain and Place Royale in the Lower Town. The Château Frontenac is the unmissable landmark — even if you're not staying there, the guided tour of its history is worthwhile.
The Plains of Abraham west of the walls are the site of the 1759 battle that ended French rule in Canada; the Citadelle on the bluff above runs daily guided tours and a Changing of the Guard in summer.
For museums: the Musée de la civilisation in the Lower Town is among the best in Canada; the Musée national des beaux-arts covers Québécois visual art from the colonial period to the present; the Ursulines Convent contains Montcalm's skull and the history of North America's first educational institution for women; the Morrin Centre is a former colonial prison turned Victorian library.
Day trips from the city: Montmorency Falls (15 minutes east, taller than Niagara), Île d'Orléans (20 minutes, farm island with excellent food), and Mont-Sainte-Anne (45 minutes, major ski area).
In winter, the Hôtel de Glace at Valcartier and the Carnaval de Québec in February are among the best cold-weather experiences in North America. In summer, the Festival d'été in early July draws major international acts to the Plains of Abraham.