There are a lot of places to see besides the beaches of north& south Goa. You can go see Chapora fort(of Dil Chahta Hai fame) , Aguada Fort, Reis Magos fort, Basilica of Bom Jesus, Our Lady of the Immaculate Conception Church. Also you can go for the famous boat Cruises in Panjim. If time permits see the salaulim dam, go for DudhSagar trek(best in monsoons).
If you are a peace lover visit south Goa, Benolim beach, its breezy,beautiful and serene.Its a very clean beach and the surroundings are also quiet. Good place to sit relax and away from hustle of city life. If you are looking for exactly opposite party music and a place filled with activity try North Goa, Anjuna beach, Agauda beach