Okay now, overcome your beach hangover and take up the hiking route. Goa has some fabulous trekking routes and the route to the Dudhsagar Waterfalls is one of them. Staying true to it's name, the waterfall does appear like a 'Sea of Milk'. The best time to visit here is in monsoons for obvious reasons. And also, during the dry season you won't be able to see it's actual beauty.
I had saved the best for the last. Well, this is probably the best offbeat thing you can try out in Goa. There are a few NGOs in Goa that give you an opportunity to volunteer. So while I was staying at the hostel, I met Sarah, who was going to head to Arambol to volunteer to help save and treat the street dogs. Though I didn't really sign up for any particular program, I got to spend a few hours with a volunteering team who had got the street kids to enjoy the beautiful Ashvem Beach.
I am planning to undertake a program through Volunteering Goa but you can definitely look up for some more options over Google.
BONUS. Watch a local football match