two castles in the Loire Valley

Jonna, a lot depends on when you are going and do you want to see them both on one tour, or one day for each?

Without a car, Tours is probably the best place to base to see both. You can reach Chenonceau by train. If you're going in July or August there is a shuttle bus from Tours to Villandry. Another option (if you'll have a couple of days) is to stay in Chenonceau and make your way to Villandry by train/bus.

There are lots of organized tours from Tours that include 2, 3, or even 4 of the chateaux in one day, but I would think only the full-day tours would go to both Chenonceau and Villandry in one day. The biggest problem with organized tours is the limited amount of time at each chateau which is sometimes not enough to satisfy. You could also arrange a private tour doing both but it will be much more costly. Or you could rent a car for one day and see them both.

I did a group tour by bus that included Chenonceau, Chambord, and Amboise (we were staying in Blois) but, as I said, not enough time at each for me to just wander around like I tend to do. I also went with a car once and stayed in Azay le Rideau and drove to Villandry.

Because these two are on opposite sides (East and West) of Tours, your plan is challenging. I think the first reply was very helpful. I will obey your #4 demand, to your own peril.
The exterior (i.e. gardens) of Villandry are better than the interiors. Distances are large, but there are woods to explore as well if you have time and it's not too hot. Both of these are busy, high traffic destinations. In summer, Villandry has a "garden produce bin" where you can take lettuce or whatever they picked that day and didn't sell or give to charity. It's on the interior wall to the "right", before you climb up steps to the Jardin des Simples. Of course you can't pick yourself from the ground.

Villandry is really easy to get to by bus from the Gare de Tours. It leaves from RIGHT out front on a yellow bus. It is a quaint, nice ride.
The easiest way to get to Chenonceau is by private transfer or train. See the nice people at the Office de Tourisme there to ask more about this. When facing the outside of the train station, if you look right it is the most modern looking building.

Breaking rule #4...because you know you're tempted:

If you're willing to see more, there's a company called Quart de Tours that provides transportation (not a tour or entry). The service is impersonal but they get you there hassle free.

Excursion 7 will get you:

Départ 9h - Retour 18h15
Azay-le-Rideau
Villandry
Déjeuner à Tours (lunch)
Amboise
Le Clos Lucé (Leonard de Vinci's house)
Chenonceau
Dégustation de vin (wine tasting)

all in one day. 50EUR + entry fees. But they do get you a group entry price.

Tours itself is an amazing city. You could stay near the train station (Le Grand Hotel next to the train station is very nice and close to the office de tourisme as well).
If you have any questions about Tours or that area, let me know. I've done my fair share of travelling and living in that region.

Try not to over- or under- estimate the time you'll spend at each place. Neither of these chateaux are very big. It's also the grounds that are the most important parts of these two.
Bon voyage!

Ohhhhh, I LOVE the last reply. I have seen all of those chateaus, and they were all worth it.

Here's a thought: Rent a car! It is SO EASY to drive in France. They drive on the same side of the road as we do, everything is well marked on the freeways, and it is easy to do. I rented a car at the train station for 5 or 7 days (can't remember how long it was) and did a driving tour of the Loire. I am a single female, I was in my late 50's with my adult daughter. Got a map at the gas station, and had no problem getting around.