I'm quite often asked (especially on YouTube) what materials I use, so the person asking can go right out and buy all the stuff at once. It's not really that simple. The materials I prefer may not be what someone else would choose, but I guess it's somewhere to start if you really have no idea about what you should get.
To soften the shapes further I put some Vaseline (petroleum jelly) on a finger, and rub it over the clay. I also have texture stamps made by applying latex or firm silicone rubber over surfaces with interesting textures, like bumpy plastic displays in cars or various fruit. I seldom use them nowadays, but these flexible texture stamps are a quick way to add interesting surface detail to your sculpture.
I often mix the latex with other materials, mostly cotton, to add volume. But it's important to remember to never add a cotton/latex mix to an area of the puppet which needs to be soft and bendable.
To create the "bones" or the hard parts of the wire armature I use Friendly Plastic thermoplastic. This material is brilliant, and I've been using it for years for all sorts of things. I've made entire sword handles from it, as it's very sturdy. You melt these plastic strips with a heat gun, and if you have to re-do something, just melt the stuff again and again.
Master animator and puppet builder Jim Danforth recommends a brand called Pliobond.
But when the base paint is dry I attack my subject with an airbrush, in my case the Iwata HP-C double action airbrush pen. I can't say if this model is still available, but it's another little tough guy, very precise and easy to clean. I use a small air compressor specifically made for airbrush work.
I've also made masks and puppet parts out of Dragon Skin FX-Pro, and there are many ways you can vary the softness of the silicone using additives. I can also recommend PlatSil Gel 10, which is another easy to use silicone ideal for both moulds and mask/puppet projects.
Sometimes I use Smooth-Cast 65D, which is more elastic, and therefore able to take a bigger beating.
Are these material expensive, then? Well, the SmoothOn products cost a bit, but are worth every penny. The Monster Clay is also quite pricey, but other than that you won't have to spend an arm and a leg tooling up for your first puppet or mask project.