Visit North Goa if you want to party hard and want to see crowded beaches. South Goa is widely known as honeymoon spot with serene and peaceful beaches.
Since its your first time, stay in north goa preferably in Anjuna/Vagator which is less crowded than Baga/Calangute area.
You can visit Casino if you are interested in gambling, one of a kind experience. Visit Anjuna flea markets on Wednesday or Arpora Night Flea market on Saturday night. Titos lane one of the best for night parties. If you want hippie crowd visit Arambol beach around 4pm in evening. You get to see some fusion music on the beach.
Rent out bike,GPS , try to stay near calangute. go to agoda.com for good and cheap stays.and just search google for things to do in goa. u won't have any issue to roam there.
Go ahead and visit South Goa as you will have somewhat privacy at the beaches and its cleaner. If you want to experience the North Goa side then do it during the December time
Goa is divided in two parts. South Goa is calmer, less touristy but very beautiful. If you are going to enjoy beaches, peace and Serenity, stay there. North Goa is vibrant, full of life and everything Goa is famous for so go there to enjoy parties. If you have time, you can do Dhudhsagar trek and other hidden trails. You can make use of free ferry services for traveling like a local. They can guide you to other good places which no outsider can do.