Vieux Lyon, Presqu'île, and Croix-Rousse are all walkable from Place Bellecour. The metro is just for longer distances and the airport.
A pleasant half-day route: Place Bellecour → Presqu'île → Vieux Lyon traboules → Fourvière funicular up → walk down through Roman ruins.
Wear good shoes; lots of cobblestones in Vieux Lyon.