FrenchMystiqueTours will probably see this post and respond, but if he doesn't, search for some of his posts on this board. Or Google the same to see the organized tour options. Have fun!

It takes two hours minimum to get to Tours from Paris, and two hours back from Tours (the major train hub). Then you have hours and hours of cycling to get out of the big city and into the châteaux country to visit (whether you want to visit châteaux or not), then hours back to Tours. This is a plan that is not going to work for a daytrip. Even if you could manage the logistics, which I doubt you could, it would certainly not be "relaxed day of cycling."

There are innumerable places much closer to Paris where you could probably rent a bike and tour around at least semi-countryside.

Nope, you can get TGVs between Paris and Tours that take just over an hour (not counting getting back and forth to the train station in Paris obviously).

I can't even ride a bike, so I'm not a world expert on this subject, but I know numerous people who have cycled from Tours to Villandry, a château mainly renowned for its gardens, in a day (admittedly not with a train from Paris tacked on either side).

It's a 21 km ride from Tours, so that's what, an hour each way (flat terrain as far as I know)??

http://www.loirevalleytourism.com/Discover/The-essentials/Weekends-discoveries/Cycling-in-Touraine.html

Here's a site I found offering cycle hire in Tours
http://www.locationdevelos.com/en/locations.php

and general information on cycling in the Loire
http://www.cycling-loire.com/

Whether or not that sounds doable or pleasurable to you, I don't know, but you can make up your own minds I'm sure

PS that cycle hire place I linked to also lets you drop your bike off at a different location from where you picked it up, so I'm sure you could work out a leisurely route including an overnight stay and then hop on an early train back to Paris if you wanted

If I were to spend a day cycling in the Loire Valley, and I do believe this is really too far from Paris just for one day´s cycling, Blois is cycling headquarters with many routes and rental companies ready to accommodate cycler´s needs:

www.bloispaysdechambord.com/en

Perhaps the easiest way to reach Blois would be taking a morning train from Gare Austerlitz, which might require a quick change in Orléans. There is a 07h38 direct train from Austerlitz which arrive in Blois at 09h00. From Blois, which has one of the area´s most interesting châteaux right in the center of town, it is a relatively short distance to Chambord, Cheverny, and you could reach Chenonceau depending upon your endurance.

There are a lot of possibilities but I might at least spend one night before returning to Paris.