It depends on what you are looking for. If you have never been to Goa before then you must first decide what you are looking for in Goa. Goa is mainly divide as North Goa and South Goa. Both the areas of Goa has different things to offer. North Goa - More tourist people,beaches, parties, flea markets, watersport activities, beautiful streets. In other words a lot more happening place. South Goa - Cleaner Beaches with less people , pretty sunsets, peaceful and quite compared to north goa. So you must decide what you are looking for - parties or relaxing holiday. I recommend if you are going to Goa first time, the visit North Goa places first
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.
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.