You can make a very nice bow using only a splitting wedge, a hammer to drive it, and a sharp machete (and some knowledge about how wood bends). You also need a cooperative tree or dead limb, some string or cordage, and about an hour and a half.
Black Locust, Red Mulberry, White Ash, Hickory
1) Find a nice, straight limb about 5 or 6 feet long, preferrably dead and standing if locust, mulberry or Osage (fence posts are great). If ash or hickory, wood cut green and dried indoors is best. Even a dried 2" thick board will work if the grain is straight and properly oriented in the board.
2) Split the stave after cutting it to length and select a piece that does not twist for the bow. Avoid knots with bad holes, splits or other obvious defects. Illustration.
3) Cut out the outline of the bow along the grain of the stave. Make the back of the bow (the side that faces the target) from the FIRST ANNUAL RING of heartwood, or from the sapwood above it if the sap is thin.
4) Mark off the handle area, then hack or carve away the excess wood on the belly of the bow. The belly is the side that faces the archer. Carve it away until you can bend the bow over your chest just a little. Make sure that it bends evenly over the length of the bow.
5) When it feels like you have a 75 pound bow carved out, whittle any kind of notch in the ends and tie some string on the bow to bend it slightly. The best bowstring material is dacron, but twisted gut, rawhide, twisted sinew or even nylon will do.
"Tiller" the bow by scraping more wood from the belly of the limbs. Hold the machete blade at a right angle to the belly and shave or scrape off ribbons of wood to weaken the working part of the limbs in a slow and controlled manner. It helps to have a vise or a flunkie to hold the bow at this point.
6) Occasionally pull on the string ( gently at first ) to ensure that the scraping is weakening the limbs equally, and stop when the poundage is about right. In the later stages a tillering board or stick can be used to hold the bow drawn while you continue to scrape. A bow gets stronger as it dries, so if you go a little too far in weakening it might work out okay.
7) Wrap a handle on the mid section so that the arrow does not slap it loudly. Cloth, leather or woven bark is fine.
8) Check your tiller at full draw to make sure the limbs bend evenly. What makes a bow work is thinning the stave evenly along it's length in order to share the burden of bending. Even the best wood cannot take being mistreated by weaking one spot too much. Take your time and be careful. A primitive bow should not be drawn more than one half of its length.
Arrows are not so easy as bows. But they are not difficult once you learn how it is done. The Native Americans pulled turkey feathers apart and then lashed the vanes to a shaft with deer sinew (tendon). Feathers were often glued into place then secured with a spiral wrap of sinew to make an arrow with fletching more durable than today's glued only arrows.
Everybody knows about stone arrowheads, but most cultures used bone and, if they could get them, metal arrowheads. Like the feathers, arrowheads were attached with animal sinew and a variety of natural glues. Arrowheads are fascinating even without an arrow attached, whether you like to make them, find them, or buy them to collect or mount.
There are several good books on the subject of arrowmaking. "Making a Plains Style Arrow is a 21 page pamphlet that deals exclusively with making arrows and has nice photos. "Bows and Arrows of the Native Americans" by Jim Hamm is a super reference on Native American archery with a long section on arrow making. Both of these publications are inexpensive.