Puppets offer many possibilities for arts integration. Language arts connections are easy to make since puppets engage in expressive dialogue; anything children can read and say, puppets can too. Similar connections can be made to other subjects as well. Jim Henson was a master puppeteer who left us this marvelous video.
As you can see from the video, there are many ways to create puppets. It is not necessary to spend a lot of money or time to come up with an effective puppet. Puppets don't have to have mouths that move either; think of Mr. Rogers and his puppets, as in the following video about Fred Rogers and Daniel Tiger.
If your puppet does have a mouth that moves, it is important to think about how it should move to make each syllable of each word. This is great practice for kids in thinking about words, syllables, and enunciation. Basically, the mouth should open (rather than close) on each syllable. Here's a video tutorial about moving a puppets mouth, courtesy of the Jim Henson Company.
For another example of a simple puppet with a mouth that moves, watch the following video of Shari Lewis with her sock puppet, Lamb Chop. Shari Lewis is a ventriloquist, so her own mouth doesn't move when her puppet talks. But, you don't have to be a ventriloquist to be an effective puppeteer.
Whether or not your puppet has a mouth that moves, you will need to develop a voice. Think of all of the things that go into the sound of a voice. The overall type of sound is often referred to as the tone quality or timbre. When you hear someone's voice and you know who it is without seeing them, that is because their voice has a distinct timbre. Words to describe timbres could include things like nasal, raspy, stuffy, etc. Other considerations include volume (how loud the voice is), tempo (how quickly the words are spoken), and pitch (high, low, or in the middle). Then, of course, mood also can make a big difference in how a particular voice sounds. Remember, too, that how the puppet moves while speaking (or not speaking) can communicate a lot. Here's a video of a puppeteer discussing how to develop puppets' voices and personas.
One more thing you may need to think about is a puppet theatre. You don't need to go to a lot of expense to create something that is effective. Basically, children will just need something to kneel behind. The size of the puppet theatre can depend on the size of the puppets. Finger puppets, for example, could perform on a stage created from a shoebox. Allow the children ample creativity in creating puppets, puppet theaters, and puppet plays or improvisations.