How can I make my own free bow and arrow in my backyard just from things in the woods?

Depends on your backyard, ideally you need a yew tree for the wood to make the bow itself, you would have to make some stone tools first in order to cut and shape the wood. Then you would need a source of hot water and a frame to bend the wood over so that when it cools it retains the shape of the frame. The frame would obviously induce a curve in the wood. Flint or chert would be ideal for that purpose. Then you need some reeds, specifically the pith which can be wetted and rubbed to form fibers and then a string for your bow. This is a skill and will take practice, animal fibres (wool) will not work, you need a woody fibre. With the arrows you need to either collect straight sticks or form the shafts from a block of wood. The latter is better is you want more than a half dozen. Small stone blades can be used for the arrow heads, usually off cuts from when you were knapping flint to make your initial tools. Flights for the arrows can be a broad leaf inserted into the shaft of the arrow, feathers glued on (making glue the glue is slightly complicated but tree sap or resin can be used in a pinch) or simply fray the end of the shaft. The method you use will determine accuracy and accuracy over range.

There are other things too and alternative methods/wood/plants that can be used but the above will give you a basic bow in a day if you have the materials to hand and the skills to make it.