It shouldn't matter that much if the mouthplate is cardboard, I think the most important thing is that your hand has a snug fit and doesn't flop around inside the head. You could use a strap or just stuff some foam or stuffing into the head, say above where you hand goes to make it a tighter fit.
The second most important thing is to only move you thumb, and keep your fingers (top half of mouth) as still as possible so the puppet has good eye contact.
I third the notion to just practice, practice, practice in front of a mirror. It takes time, but once you feel comfortable then a private lesson or two would be a good next step. That's where I am right now....
p.s. Thanks Buck for the link to Puppetry Lab - i had been looking for that for months and thought it just disappeared. I loved the videos, and now there is great writing as well.