Before deciding on the materials to use for a rug, consider three important factors: durability, care. and appearance.
There is hardly any limit to materials that can be used to make a rag rug. Rag rugs were made originally from materials that were readily available, be this woolen blankets, sheets, shirts, or coats, no longer wearable, but still containing good material. More delicate rugs would be made from more sophisticate materials, such as silk or shantung. Some materials will wear better than others. Generally polyester does not wear that well, so do not expect a long life from a polyester rug. Natural materials, such as wool, linen or cotton will give beautiful results.
Blankets and coats were often made from wool. Wool is in a category of its own with some good qualities and some problematic ones. Wool will absorb water to a great degree, but of course needs to be carefully washed or dried cleaned to stay soft. Although wool cannot be washed in a washing machine in the ordinary way, it can be done in a top-loading machine. Wool can absorb a great deal of water. A saturated rug becomes very heavy, but can be spun dry. To prevent the wool from matting or felting, avoid friction. With care small woolen rugs can be soaked in the washing machine, using cold water. After soaking for 15 minutes, let the machine do a few rotations while you stand there (three to five seconds), and then proceed to the spin cycle. Then let the machine fill for the rinse cycle, stop the machine after filling, and then let it again do a few rotations before proceeding to the last spin cycle, making sure the material is carefully spread against the outside of the drum to prevent it from stretching. When spun dry, roll in towels to remove more moisture and then lay flat to dry, away from direct heat.
Surprisingly enough this same quality of matting or felting can be used purposely when a less resilient surface is desirable. This can be done professionally as well.
For durability, use wool, cotton, or linen. For ease of care, use cotton, linen, or synthetic materials, such as acrylics. Leather and flannel can also be used. Choose from upholstery fabrics, drapery materials, and blankets.
When you run out of materials to work with, consider visiting a second-hand store or thrift shop to buy drapes, coats, and blankets. This will allow you to use inexpensive materials to learn the craft.

