Here are basic materials for getting started:
- Sewing machine. I have a Brother sewing machine that I bought from Costco ten years ago. I used it to sew my wedding dress, a couple of jackets, and many other random projects and it still works great. I dream of owning a Serger - someday...
 - Good fabric shears. Otherwise you will give up before you get your first outfit cut out. Mine are Gingher brand, very similar to these
 - Pincushion with pins
 - Seam ripper
 - Needles
 - Thread
 - Something to store all of your sewing supplies in. I use a fish tackle box my Grandma gave me to use as a sewing box the Christmas I was six years old.
 
Extras you will probably want to get eventually:
- Buttons
 - Elastic
 - Zippers (my daughter won't wear anything with a zipper in it, so if your child is the same way, don't bother)
 - Embroidery scissors - nice for cutting thread but so are the child safety scissors you have for craft time.
 
A few tips:
- Don't be afraid of buttons. Most modern sewing machines have a buttonhole feature that makes sewing buttonholes very easy.
 - Straight pins are your friend - it's well worth the extra time it takes to put them in your fabric to keep seams straight and line up.
 - Zig-zag edges of woven (non-stretchy) fabrics before sewing together to prevent fraying
 - Use a zig-zag stitch when making clothing out of stretchy fabrics. A straight stitch is fine for wovens (if a woven item is cut on the bias - which makes it a bit stretchy - you probably want to use a zig-zag stitch).
 - As a general rule, pattern pieces for most articles of clothing should be cut out in such a way that the vertical line of the clothing (when the wearer is standing up with their arms hanging at their sides runs parallel to the selvage.
 
If you're looking for a book, The Reader's Digest Complete Guide to Sewing : Step-By-Step Techniques for Making Clothes and Home Furnishings is the sewing reference I turn to most often. You can get used copies off of Amazon for almost nothing. 
Have any questions, comments, or other tips? Please let me know!