Local buses are an option in some areas. Transco runs buses from Dijon to Beaune and some of the neighbouring wine towns, and also serves destinations north and west such as Châtillon-sur-Seine and Semur-en-Auxois.
Regular trains (more expensive but much faster than the bus) run between Beaune and Dijon. There's also frequent train service from Dijon northwest along the TGV line to Montbard.
To really explore the Côte d'Or vineyards and other more rural destinations, your best transport options are car or bicycle.