Dudsagar Waterfalls: a lovely drive to the waterfalls, and when you arrive, you get into a jeep and drive up over rocks through the river, and then take a 15-minute walk over the rocks until you reach the magnificant waterfall. Buy bananas and nuts to feed the monkeys, but don't leave anything lying there and go for a swim under the waterfalls.
Coco beach is lovely as well; all taxi drivers know where it is.
Go on a boat trip from Coco beach and see the dolphins and Goa's only jail. The boat guys will call out and you might see some of the prisoners.
Come to the Saturday night market in Arpora for live music, food, and lots to buy, or to just chill with a beer.
Quick Tips:
You should try the beach shacks for eating; we go to the Lucky Star along Calangute beach.
Also have all your beauty treatments (manicures, pedicures, massages) here because it is less than half price than at back home.
When buying, haggle; don't be scared because that is part of the Goa experience.
Best Way To Get Around:
The best way to get around is by the local taxis, a scooter/motorbike for 150/250rs per day, or just walking.