Gran Paradiso is one of Italy's best walking areas, with over 700km of trails linked by a recuperative network of rifugi (mountain huts). The tourist office has free winter and summer trail maps for walkers and skiers.
The main point of departure for the Gran Paradiso peak is Pont in the Valsavarenche. Technically it's no Mont Blanc and can be completed in a day, but you'll need a guide (a two-day ascent for two people starts at €500, less for larger groups). Contact the Società Guide Alpine di Cogne.
Despite its wilderness credentials, there are in fact 9km of downhill slopes here. A one-day ski pass covering the use of Cogne's single cable car, chairlift and drag lift costs €25. But most winter visitors come here for the 80km of well-marked cross-country skiing trails (per day €5) that line the unspoilt Valle di Cogne. Try trail 23 up to Valnontey and Vermiana, or head east to Lillaz. Skiing lessons are offered by the Società Guide Alpine di Cogne. They also offer ice-climbing expeditions on the Lillaz waterfall.
Easy walks around Cogne include the 3km stroll (wheelchair accessible) to the village of Lillaz on trail 23, where there is a geological park and a waterfall that drops 150m in three stages. Trails 22 and 23 will also get you to the village of Valnontey, where you can continue up the valley to the hamlet of Vermiana (1½ hours one way). Trail 8 from Cogne leads to another waterfall (Pila) via the village of Gimillian.
A classic, moderately strenuous hike from Valnontey is to the Rifugio Sella, a former hunting lodge of King Vittorio Emanuele II. From the town bridge, follow the Alta Via 2 uphill for two to 2½ hours. More adventurous hikers can continue along the exhilarating Sella–Herbetet Traverse (15km), a seven-hour loop that will drop you back in Valnontey. You'll need a head for heights and a good map.