Here is an illustrated guide on how to build a wooden tree swing.
Tree and branch: What kind of tree branch? You want a branch that is strong enough to hold the weight of an adult. You want a branch that is far from the trunk so that the swing will not hit the trunk. You want a branch that is not close to any other branches so the rope won't hit anything. Prune if necessary. You want a branch that is high enough for a swing. I picked an evergreen tree in our back yard. The branch has about about an 8 inch diameter, and is about 15 feet from the ground.
Rope: Nylon rope is good because it holds up in weather. Pictured is 100ft of 3/8 inch nylon rope (240 pound test). Since the swing will have two ropes, that gives you 480 pounds of strength. About $10.
Bench: A piece of 2x6 wood for the bench. I had a scrap piece that was about 20 inches long. Drill holes in the four corners with a 3/8 drill bit. Drill close to the edge, but not so close that the wood is weakened. The holes should be wide enough for the rope to go through, but no wider. Sand the wood so your kid doesn't get a sliver in the behind. And apply wood finish to help minimize weathering.
Eye bolt: Get two heavy duty eye bolts with washers and nuts. Drill holes through tree branch (you will need a 3/8 foot long drill bit). Run bolts through the branch. The ropes will attach to these. DO NOT loop the rope over the branches. The rope loops can fray and weaken the rope, and the wear on the bark can kill the branch. Buy strong bolts that will support the weight. About $8 per bolt.
Bench Rope: Cut two 4 foot lengths of rope, put them through the bench drill holes, tie simple knots and burn the ends so they don't fray. My lovely thirteen year old demonstrates.
Swing Ropes: Cut two long pieces of rope for the swing. Tie loop knots on one end, and run the rope through the eye bolt and loop. Burn the ends so that they do not fray. You can see this step in the first picture with me on the ladder.
Attach the swing: Attach the swing and you're done! You can do a fishing version of the loop knot called the "Palomar Knot" (description here) to attach the wood. The four foot loop of rope from the swing will be your fish hook. My 5 year old and 9 year old demonstrate.