For goa check arambol, mandrem, Betalbatim, cola, agonda, cabo derama, palolem, butterfly, galgibagha, xandrem etc. For gokarna all beaches are good and within small distance, om Beach, kudle, half moon beach, paradise beach etc.
In Goa: Arambol for shoping, beauty, tatoo and good food
Cola peaceful Depends on the kind of taste and also one can travel along the beach side to explore a better calm location.
All the beaches are great in both these places but to visit all the beaches in Goa, you might need to stay for little longer, covering both south and north Goa beaches would need some time