Travelling to Chateaus in Loire Valley without car

Tricky - the chateau you have tickets for are all in different directions. The only two you could comfortably do in a day without backtracking are Chenonceau and Amboise. You'd have to catch the early train to Chenonceau (about 9.15 if my memory serves me correctly) then the 12.30 bus to Amboise. From Amboise you have a choice of trains back to Tours.

Chambord will take you all day - train to Blois, shuttle bus to Chambord.

Villandry ifs theh other side of Tours to the three other chateau. Depending on yoru dates there may (or may not) be a bus.

Train times are here https://www.horairetrain.fr/
The Blois shuttle info is here https://www.remi-centrevaldeloire.fr/loir-et-cher/navettes-chateaux/
Bus times for Chenonceau/Amboise and Tours/Villandry are here https://www.remi-centrevaldeloire.fr/itineraire/

I have visited all those chateaus, and I my opinion, those are great choices. Clos Luce' because of the daVinci connection, Chenonceau for many reasons (!), Villandry for the gardens and Chambord because its huge and the exterior is pretty/interesting. At Chambord you can also walk around on the roof which was interesting.

I only saw the river from Chenonceau, but in my opinion, if you can't do the boat, you wouldn't miss much. You can get nice pics of the chateau with the water in it from different vantage points. In regards to Close Luce, I would suggest allocating a fair amount of time for the interesting and pretty gardens. Also, I do think Amboise is a pretty little town and you would want some time to walk around it and also to view the Loire river.

I'm sorry that I cannot help you with the transportation piece. We rented a car for Loire. I think you might be able to take public transportation from Tours to Amboise. My recollection is that there is a train route.

You do not need a car to accomplish what you want to see.
This is how I would do it:

Day 1 take the bus number 117 to Villandry. You can catch in front of the Gare de Tours trains station. It leaves at 08:51 and takes 30 minutes. You can pay 2 euros on board. You can use the same bus back to Tours direct leaving at 12:41.
Here is the schedule
https://www.filbleu.fr/images/infosvoyageurs/ficheslignes/reseau_ete_2019/117_Ete2019.pdf

Day 2 Take the TER train to Blois. At Blois in front of the train station you catch the bus to Chambord
Here is the schedule
https://cdn1.chambord.org/en/wp-content/uploads/sites/3/2019/04/FLYER-HORAIRES-REMI-NAVETTE-CHATEAUX-2019.pdf

Day 3 Take the earliest TER trains to Chenonceaux visit and then take the train Amboise and then take the train back to Tours. Amboise is a small town

Not sure about "Day 3 Take the earliest TER trains to Chenonceaux visit and then take the train Amboise and then take the train back to Tours. Amboise is a small town"

Amboise and Chenonceaux are on different train lines. From memory the fastest connection by train is about an hour - the bus is about 20 minutes

FYI, when I was in the Loire about 4-5 years ago, there wasn't train service in Amboise on a Sunday, or at least between Amboise and Tours

Thank you guys for your response.

Do we have to books train/bus tickets between the castles in advance? Just like how we book our plane tickets in advance.

Thank you so much.

No need to buy tickets in advance for the local trains, and there isn't even a way to do so for the buses.

We stayed in Amboise during the summer without a car. The TI in Amboise has information/reservations for shuttles to nearby chateaux and they can also help you with local bus schedules. We took the local city bus to Chenonceau which is a very easy direct bus that drops you off right at Chenonceau. Another day, we used the Navette Chateaux excursion bus (found it in RS guidebook) that leaves from the Blois train station: 6 euros for a day pass that stops at 4 chateaux plus you receive admission discounts to the chateaux. We loved the Loire Valley.