The ancient city of Anuradhapura has Stupas and temples scattered around the city. It is lush green, and decorated with gardens along all the roads. Things that a Google Search won't tell you :
1. Wear clothing that covers your knees ,and shoulders. If you want to fit in the culture perfectly, adorn white shirts with skirts/dhotis .
2. You may choose to buy a few lillies and arrange them at the prime spots inside the temple.
3. Temples with bodhi trees have a custom of circumscribing the trunk thrice with a cauldron of water in your hands. After circumscribing, you have to pour down the water at the roots of the tree.
Not to be religious, but, following these customs is quiet an peaceful experience in itself to fully understand the local culture.