I use dowels for arrows all the time.
Here's what I'd do:
Birch dowels...5/16" should work.
Buy more than you think you'll need.
Check to make sure the grain is as straight as possible.
Cut them about 4" longer than your draw length. For many people, the longer the arrow, the easier it will be to hit the target.
Cut simple knocks into the dowels, sharpen the tips with a knife, shoot at your target from about 10 feet away, and pick the ones that fly pretty good without feathers.
Add feathers and points to the ones that fly the best. You'll probably have 3 good dowels for every 10 that you buy.
Stores that carry "hobby" stuff usually have birch dowels....near the balsa wood section.