Hampi and Gokarna although around 300kms apart are on same route from central to north Karnataka while Coorg (Kodagu) is exactly opposite. All three places are different in nature. Hampi is a archaeological UNESCO heritage site, Coorg is a hill station with coffee and spices plantation and Gokarna is a quiet beach place different to Goa. You can cover Mysore with Coorg if you haven't been there yet as it will come on the way. Honestly 6 days is too less to travel as most of it is going to be road trip and you will have to travel through entire length of the state.Try to travel during night both to save time so you can see places during the day.
Coorg: You can take overnight bus and can stay in hostel/home stays. You will love local food in Coorg specially if you are non-vegetarian.Bangalore to Coorg is around 6.5 hours by road.
Hampi: There is a direct overnight train from Bangalore to Hospet (near Hampi). Homestays are easily available and vegetarian food is served. Try to stay in Hippie island on other side of river. You might like it.
Gokarna: Direct bus from Hampi. Some 7 hours i think.
Note that there are direct buses available between each of these locations and Bangalore Bangalore to Hampi is best by train so you can sleep but other routes are best traveled by bus as train journey would take more time and may not be direct. Let me know if you need any more information or help. Welcome to Karnataka!