There are many places in Karnataka to visit in the month December. See if you are looking out to witness mist and feel the nature you better visit Chikmagalur, Coorg or Sakleshpura.
Coorg and Chikmagalur for tea estates, tea plantations and beautiful waterfalls Hampi for its beautiful architecture
Gokarna for beach vibes
Murudeshwar for both beach and temples
Udupi for temples and beaches
Mangalore for food and beach