Three days split cleanly here: a full day in Montreal, the transfer morning, an afternoon and evening inside the old town, then a whole final day out at the falls and the island. This is the order I would follow.
Day 1: Mount Royal and the Olympic Park Museums
Give Montreal the first day and start it on the mountain. Mount Royal Park rises above the downtown grid, and the walk up takes under an hour at an easy pace. Saint Joseph's Oratory of Mount Royal, the domed church on the same hillside, charges $6 for an adult and earns its hour.
Spend the afternoon around Montreal Olympic Park, where the Biodome is the headline stop. Three hours covers the animal habitats, and the botanical garden and the Insectarium usually fit into the same half day. The Insectarium runs its visits from November through April, so check the calendar if the bugs are the reason you came.
Dinner back downtown, then an early night. Tomorrow starts with a train.

Day 2: Train to the Old Town
Take a morning departure, and you will be in Quebec City by lunchtime. Drop the bags, then walk the Upper Town: the Château Frontenac, the hotel the Canadian Pacific Railway built in the 19th century, the Dufferin Terrace beside it with its river view, and the Citadelle above. The walls around them are why the old town carries UNESCO World Heritage status, and an hour on that walk explains it better than any plaque.
The afternoon belongs to Lower Town. Ride the funicular down, wander Place Royale, and work through the lanes of Quartier Petit Champlain, the oldest commercial district in North America. When the funicular queue looks long, the Breakneck Stairs do the same job in ten minutes.
Climb back up for the Plains of Abraham, the old battlefield that locals now use as a park, then eat dinner on Rue Saint-Jean. If rain settles in, the Musée de la civilisation and the Musée national des beaux-arts du Québec each hold a full half day indoors.

Day 3: Montmorency Falls and Île d'Orléans
Head out of the city in the morning for Montmorency Falls. It runs 83 meters high, roughly 30 meters taller than Niagara, and it sits a short drive from the walls. Ride the cable car up, walk across the suspension bridge at the crest, or take the zipline across the falls for a closer look. Two hours is plenty.
Spend the rest of the day on Île d'Orléans, where a bike is the best way to see the island and the loop passes farms, a cheese shop, and a cider producer. It is the slowest stretch of the three days and the one I would protect if the sun shows up. If the weather turns, save the island for a drier day rather than force it.
Come back into the city for a last dinner in Saint-Roch, east of the old town, the neighborhood with the livelier food scene.
