The best time to visit is June through August, when the days are very long (Petrozavodsk is close to the Arctic Circle, and summer nights barely get dark), the hydrofoils to Kizhi run frequently, and the outdoor activities in the Karelian countryside are all accessible. July is the peak of summer and the busiest month.
Winter visits are for the hardy: temperatures regularly drop below -20°C, the days are very short, and the hydrofoils to Kizhi don't run. But the frozen lake is beautiful, cross-country skiing is excellent in the forests around the city, and the relative lack of tourists makes the city feel more authentically itself.
Late May and September can be pleasant — cooler, quieter, and the autumn colours in the birch forests are exceptional.