Not a local, but from a recent visit I recommend:
- public skating at Place d'Youville (beside the walls of the old city) - or - on the Plains of Abraham;
- Rue St. Jean for walking, colorful shops (esp. Moisan, the oldest grocery store in N. America http://jamoisan.com/en/epicerie/lepicerie/);
- walk down the steps behind the Chateau Frontenac, and stroll the cobblestone streets around Rue. St. Pierre;
- the ferry to Levis for fabulous nighttime view of the city;
- the farmer's market next to the train station.