The Habarana and Sigiriya area does this best. A morning typically involves a bullock cart across paddy land, a catamaran across a tank, and lunch cooked over firewood in a village kitchen — several curries, a coconut sambol pounded in front of you, rice from the field outside. You do the cooking as much as you want to. It is unpretentious and the food is usually better than anything in your hotel.
Elsewhere: stilt fishing on the south coast between Koggala and Weligama, which is now largely performed for visitors but is still worth seeing when arranged with the actual fishing families; the Tuesday market at Dambulla, the largest wholesale produce market on the island; cinnamon peeling around Mirissa, where the bark is stripped and rolled by hand; mask carving at Ambalangoda; and the batik and pottery workshops around Kandy.
These fit easily as half-days inside a larger itinerary and are usually what children remember afterwards. They also happen to be the cheapest part of a trip and the part where the most money stays local.