Your child will do a lot of writing practice, so she'll need many pencils. Erasers are also necessary for correcting mistakes. The large pink or white erasers work better than the novelty ones. In kindergarten, students will typically not be using pens, so don't buy them.
You'll need to purchase a number of art supplies to get your child through the year. Crayons and markers allow for drawing pictures. Most schools also require each child to have his own glue. Scissors are another important art supply---purchase metal ones as the plastic ones make cutting difficult.
Some schools require parents to chip in on classroom supplies. This could include a box of tissues or hand sanitizers. Even if it's not a requirement, you could help the teacher out by purchasing acrylic paints or construction paper that she can use in art activities.
Your child will do better if he's able to keep her supplies organized. Purchase a box for him to put all of his art supplies in and another one for writing supplies. A backpack is essential for transporting things to and from the school. If you'll be sending your kindergartner with a lunch, you can also purchase a good reusable lunch container.
Depending on the school, you may need to purchase a small mat for nap or quiet time. If your child will be purchasing her lunch---or just milk---at the school, a small change purse can hold the money. A three-ring binder can also be helpful in keeping assignments organized. Additionally, you may want to include a separate small bag of clothes in case of any accidents.