Hi, Erika, and Welcome to APUG!
If you are looking for a camera that uses peel-apart film, you have two options. The first are the folding cameras, such as the Automatic 100 and Automatic 430. These cameras have a bellows that collapses in order to make the camera smaller for storage. The other type are the rigid plastic cameras, such as the Colorpack II. These are cheaper, but are a lot bigger to carry around. One thing to note is that none of these cameras are point and shoot; they have to be focused before taking the picture. It's really easy to do; on most of them you just slide a bar until a double image comes together.
Personally, I recommend the Polaroid 430. The camera is relatively cheap (around $20), but it has a glass lens and folds up. For a battery, you can use a CR123 battery and tape the contacts to the battery. For film, you can use both Fuji FP-100C for color and FP-3000B for black and white. These run about $10 for 10 shots.