I think it's hard to do day trips from Goa (or from most places) but I hear Hampi and Gokarna are nice. Definitely recommend Kerala too, although I guess it depends on when you are there. :)

I have traveled India 9 times and seen more then most Indians see... Amritsar, Amirnath, Kumbh Mela, Poona, Goa, Jaipur, etc...

Nearby Goa, just a few hours south is a wonderful park/island resort where you can stay in huts called Karwar. There is an almost empty beach and much peace...

More then a Day trip... try Mangalore, it's south along the coast, also Manipal and Udipi are sweet villages. In Udipi there are many Sandelwood carvings... Not as much as in Mysore, which is a bit farther south and inland. Mysore also has a cool palace to explore...

Also on my list is a visit to Bangalore (now Banglaru) which is an amazing city, with lots of parks, and a great place to shop. It's not bad for a city and the coolest discos and internet cafes...

Have a nice trip!

Please visit Kerala its Awesome!!!!