The Old City is best explored on foot — everything inside the walls is within 15 minutes' walk, and the streets are narrow enough that cars are an impediment rather than a convenience. Wear comfortable shoes; the cobblestones and steep grades of the Old City are hard on feet.
Within the Old City: Walking is the only sensible option. The Funiculaire connects Upper and Lower Towns for a small fare; the Escalier Casse-Cou (Breakneck Stairs) is the free alternative.
Beyond the walls: RTC (Réseau de transport de la Capitale) operates the bus network. Route 800 goes to Montmorency Falls and Île d'Orléans junction. Taxis are abundant and metered. Uber operates in the city. Car rental is available at the airport and at several downtown locations.
In winter: The Old City is walkable in any weather if you dress properly — boots with grip, layers, and a warm hat are essential when temperatures drop below -20°C. The city provides underground connections between some key buildings, but walking outside is generally fine. The Christmas and Carnaval seasons bring heavy foot traffic that makes the streets more navigable than you might expect.