Hey! Welcome to the awesomeness that is stop-motion!
I wouldn't actually recommend animating on a carpet unless you're going for a specific effect, the only way I can see you being able to animate on carpet would be something best saved for people with a little more experience, like the other members on here, not me though haha.
I started with Play-doh, but that is definitely not recommended, try getting some cheep Plasticine, or if you're actually more interested in animating toys then go for it. You could also try Lego if you have some lying around as Lego figures only have a few points of articulation, which is great for beginners to learn with.
I'm not the best person to give advice to newbies, but I think I know enough to tell someone how to get started haha..
I started with some small action figures about 2 or 3 inches tall. Then, I moved on to clay. I like polymer clays. You can probably find Sculpey III in packs of a dozen or more 2 oz blocks of different colors. I think this clay would be a good one for starters. Marc Spess on his website has a lot of good information. Animate Clay . Don't put your action figures or puppets on carpet! You need a table. One thing that you will learn the hard way is to have everything tied down. If you don't, then when you bump the table things move and all your work is spoiled. You need some way to secure your puppets to the table. Some people use plywood for the table top and they drill holes in the plywood and put bolts up through the holes and into the feet of the puppet. Some people use 'rigs'. This can be simply some wire that holds the puppet in place. The only problem you would have as a beginner with a rig is that you have to use some type of paint program to erase the rig wire from the photo. This is time consuming. Some people put some metal in their puppets feet and put magnets under their table. The table top has to be quite thin for the magnets to hold the puppets feet in place. Since you are just starting, then you will probably be using a webcam to take your pics. This is the easiest route.
Or you could always stick the feet down with blutack or some kind of sticky putty like substance, which would probably be the best thing to do for a beginner. :)
I started with plasticine, gave it up, and then re-started years later by making my own puppets with aluminium wire armatures. But there are a few articulated toys that some people use, and there is a lot that plasticine can do that noting else can match.
The usual way to keep puppets standing up, and even being able to walk so they still stay up even though they are off balance, is to use tie-downs. The usual method is to drill holes in the floor of your set, and poke bolts up from below. You have threaded holes, or nuts built into the feet of the puppet.
Here is a video showing how my tiedowns work: http://www.youtube.com/watch?v=jK1tAh_kCZE
You don't have to make the T and slot type that I use, they are more work to make, but it's quicker to get them into the foot while animating than lining up the threads and screwing in.
An alternative tiedown method uses powerful rare earth magnets under a steel floor, with steel plates in the puppet feet, so you don't have to drill holes.
For running and jumping, where the puppet has both feet off the ground, you need a flying rig. This can be as simple as a few strands of armature wire twisted together. After you shoot, you can erase the wires in Photoshop, TV Paint, or other paint program with layers. Some animators use the rig for ordinary walking as well - not quite as good because the foot that is supposed to be holding the weight can skate around if it isn't attached to the ground, but it can be made to work. Especially for action figures which are not designed to support their own weight by one foot, holding them up with a rig may be the best way to go - I think that's what Patrick Boivin (search him on Youtube) does for his action figure battles.
I sometimes have a set which is carpeted, but since I am working in 1:6 scale, real carpet would be much too thick and coarse. So I use velveteen or chenille fabrics. I still have a hole drilled in the particle board floor underneath, but I cut small slits in the "carpet" over the hole, so the tiedown can go through it, and it will close up again when the tiedown is removed.
Possibly this could work with real carpet - but not the carpet on your floor! You would have to have a table with a top you can drill holes into. If you want to see toys walking around your room on the actual floor, you probably need a support rig that can move along with them. ( I did that for a flying creature by having a flat piece of plywood that could slide around, and the wires were attached to that, then to the puppet at the other end.) Or you could make them drive vehicles!
I can vouch for clay models, aside from the fact that it's the only material I know inside and out.
There are two things that will help you when animating with clay, other than having a good wire armature in there. 1 is paraffin wax, which prevents the color from seeping out of the clay (you melt this into it with a double boiler).
The other is heat. Paraffin is brittle, but a remarkable thing happens when it gets heated up- it becomes flexible! If you have ever eaten that candy that looks like wax bottles with juice inside, then you have encountered paraffin wax. It will fall apart if it gets too hot, but running it under warm water will soften it enough and make it rubbery enough to let you play with it like silly putty. If you animate clay under hot lights, paraffin is your friend because it will compensate for the way heat makes clay into sludge (higher melting point). It will also make the clay shiny, so you might want a polarizing filter on your lens. This will minimize reflections and make it appear less shiny.You can also use corn starch baby powder in the double boiler to make the harder clay look a little more matte. It takes a bit of experimenting... I'm writing a book on the subject with more exact formulas, if you're interested.