The core is the Cultural Triangle, the area bounded by Anuradhapura, Polonnaruwa and Kandy. Sigiriya, the fifth-century rock citadel with its frescoes and mirror wall, is the single most photographed site on the island and deserves an early start. Dambulla's cave temples hold roughly one hundred and fifty Buddha images under painted rock ceilings. Polonnaruwa, the medieval capital, is compact enough to see by bicycle. Anuradhapura is vast, sacred and best approached with a guide who can explain what you are looking at.
Kandy anchors the southern end. The Temple of the Sacred Tooth Relic is the most important Buddhist shrine in the country, and evening puja is the time to see it as a place of worship rather than a monument. If your dates fall in July or August, the Esala Perahera procession is one of the great religious pageants in Asia.
We run these itineraries with licensed guides, not just drivers. The sites reward explanation, and the difference between a good and a bad cultural tour of Sri Lanka is almost entirely the person standing next to you.