Having been to all 6 chateaux Rick recommends...
Chenonceau and Chambord are must sees in my book. Chenonceau is arguably the most beautiful chateau in the valley and has some amazing gardens to boot. Chambord has a less impressive interior but they’ve done a lot to make it interesting (definitely get the audioguide). Chambord is also the largest chateau in the valley and you should see it just for the sheer spectacle. Definitely go to the roof!
My personal favorite is Azay Le Rideau. It’s a little fairy tale like chateau on a river. There were some interesting medieval fantasy type exhibits within when I visited. Because it’s small, it doesn’t take too long to go through either. Be sure to go up in the attic! The town that abuts it is also cute and good for an afternoon lunch stop. It’s the background of my phone, I loved being there that much.
Cheverny is a lot of fun. It is a bit more modern (modern being like 1700s) but was amazingly decked out for Easter and also had these really cool LEGO exhibits based on Aesop’s fables when we were there. I’m not sure what it looks like on a regular basis but the dogs are fun to see too. There are some nice gardens and decent cafe in the Orangerie as well.
Villandry is wonderful if you love gardens. The castle itself is just ok. Some rooms are nicely done. But my dad and I spent an entire morning walking in the gardens. It was also a much quieter place overall and was very peaceful.
Chamont Sur Loire is another one that had better gardens. The castle itself is mediocre (more so than Villandry) but we were there during the international garden festival. There were some really cool modern gardens and art exhibits scattered throughout the grounds. I actually wish we’d gotten more time there from that perspective and in retrospect, we could have totally skipped the castle for more time in the gardens. The best thing about the castle itself was view of the Loire River.
I’ve heard very good things about Chinon and the. Clos Luce in Amboise. It is is where Da Vinci spent his final years and is a bit different from other chateuax. We simply ran out of time!
So in conclusion... I would still definitely include Chenonceau (get there early!!!!) and Chambord. I’d personally try hard to see Cheverny and Azay Le Rideau. The other two... how much do you like plants? I’m sure there will be some varied opinions here.