I don't have recommendations for a cooking class.
Of course, ideas for activities all depends on your interests. Obviously, there is swimming at the beaches. We spent our first day exploring Naxos Town, eating and shopping and taking lots of photos. Our second day we rented a car for the day and explored the island; visited mountain villages and fishing villages and had one of the best fresh seafood meals of our trip. We planned on a beach day for our 3rd day but it was drizzly and cool so we weren't able to do that. Instead, we explored Naxos Town some more, enjoyed all the nooks and crannies and narrow lanes and did some more shopping for our grandchildren.
I highly recommend purchasing some guidebooks, such as Michelin Green Guide and Lonely Planet. I know there are hiking trails on Naxos, which we didn't have time to do but I assume your young active group would like to do. And we are also older, in our 70's. So your younger folks might not be content with wandering around Naxos Town for 2 days. Hopefully, you will have beach weather every day unlike us. We were there in May. June might have less rain.