you can get 3/8" dowels from either Lowes or Home Depot, or any fairly large hardware place around you area.
Use Poplar dowels. Stay away from red oak. Poplar is easier to work with to start.
All you need is a drill and some sandpaper, and you're in business. Get a sponge or some kind of pad to back the sandpaper, because it's gonna get hot with that drill spinning it. Use the drill to reduce the size of the dowel, but only how much you need for an arrow.
Once you've reduced the size a little, take some 300 grit sandpaper and run it up and down the shaft until you have a smile on your face, ot it's smooth whichever you like.
You didn't say how strong your bow is, so I'm gonna take a shot and say less than 40 lbs. I haven't found too many 5/16" shafts I liked. 3/8" seems to be the ticket. they will work great.
spend your money on feathers, nocks, and points, and keep up with who does what on this site, or one like it.
There ya go.