I'm interested in recommendations for day tours to Hadrian's Villa and the Appian Way.
A number of companies offer combination tours to Hadrian's Villa and the gardens at Villa d’Este, the easiest way to find them is to Google Viator, look through the tours and find one that appeals to you and then contact the tour company directly. The Appian Way is comparatively easy to do yourself; On Sundays it is closed to cars so mostly bicycles, the odd residents’ car and pedestrians to contend with.
Taxis and buses still run on Sunday but the vast majority of vehicular traffic is prohibited. We always have had good luck is the TI when we arrive in a new town.