I have done night trekking on vagator beach with my friends , on a full moon day , up till the fort gates . By the sunrise you can trek down. It was one of the best experiences of my life.
Morjim beach to the extreme north and Palolem and Butterfly beach in the extreme south are secluded beaches that you can visit. Along with that if you like trekking you can checkout dudhsagar falls that's nearby