The regional rail line along the riverbank will get you to the main towns (Angers, Saumur, Tours, Blois, Orleans, along with 10 other towns), while some other châteaux are served by branch lines (Chenonceau, Azay, Langeais, Chinon, along with 20 or so other towns) or SNCF bus.
Occasionally, you can arrive at the rail station and need to invest in a taxi ride to get to the châteaux buried deep in the countryside.
The station staff can recommend the best regional taxi services, many of whom have advertisements at the stations.
The downside of train travel is having to fit your visits within the constraints of a railroad timetable.
For some, that makes a hired car a particularly practical option (although minimal signage in this rural area can turn a half-hour trip into a two-hour ordeal).
The N152, hugging the riverbank, is the region's four-wheeled backbone.
Given the region's flattish terrain, hiring a bike may well appeal. There are also local bus services and coach excursions, notably from Tours.
If you're driving down from Paris, take the A10 to Orléans/Tours, or the A11 to Angers/Saumur.