In order to begin weaving baskets, you'll want to find a pattern that works for you, learn a little technique and know what materials you'll need. Traditional American baskets were made of hardwoods like hickory, ash, and oak. Because these materials are expensive and hard to work with, contemporary basket weavers use woods from the tropical calamus rotang vine. Wood from the interior of the vine is called reed and the bark of the tree is cane. Both are readily available online and some may be available in larger craft shops. Cane is also used for making the caned backs and bottoms of chairs. Reed and cane both need to be soaked in warm water for about 20 minutes before use to make them soft and pliable enough for use. Make sure that you don't leave the materials in water for an extended period of time as this will damage the reed and cane.
Choose a simple pattern to start. Good patterns are available online or in books. Make sure your pattern has lots of illustrations because many of the directions are hard to understand if just written out. You really need to see it to understand it. Start with a round, square, or melon shaped basket. If you want a basket with a handle, choose a rectangular basket with a "D" handle. For best results, you might consider ordering a kit that includes full illustrated instructions and all necessary materials.
Spokes, staves, and ribs are the basic structure of your basket and are usually added first. Weavers are the pieces that you weave in and out of the spokes, staves, and ribs. The simplest baskets use plain weave, which is a basic over and under weave. This is often called basketweave. Getting even tension is an important part of basket weaving. As you progress, there are many more complicated weaving structures you can learn and use.
For a simple basket, you can buy coiling material at most craft stores. This is often just a thick, firm, heavy string or a heavy brown paper core. Cut fabric into narrow strips, and wrap it smootly around the coiling material making sure to cover the core completely. Starting at the center of the bottom, wrap the coil in a circle and use a needle and thread to take a stitch every half-inch or so to secure each successive coil to the one before. Continue to do this, shaping the basket as you go, and soon you will have an attractive small basket.