Laminating a bow is going to be far, far more work than you want. Make a self-bow first, and then decide whether you are sufficiently besotted.
You want to start with a stave that you can bend to somewhere around 45 degrees, but that returns to straight very quickly. If you're not trying to make an awesome bow, the wood isn't too hard to find. It must be at least a reasonable approximation of straight, should not have very many offshoots (no large ones if at all possible), and I suggest you make it about 5' long.
I've seen carving a bow referred to as 'tillering' quite a lot; to do this, start by stripping off all the shoots and all the bark on the 'belly' (side of the bow that faces you). Leave the bark on the 'back'; it provides additional strength. You then want to shave wood off the belly to make it slope towards the tips; the tips should be the narrowest part. Periodically, bend the bow around your feet/a post/something or get a friend to do it, such that you can pull on the ends whilst it's supported in the middle and look at the curve. It is generally accepted that a modern bow should have a roughly D-shaped cross section; The tips should bend more than the middle.
When you are happy with the tiller of your bow, you can either carve nocks into the tips themselves (if there's enough wood) or you can make nocks from short pieces of tubing; historically, horn nocks were used. Your nocks should slope toward the middle of the bow, so that the string doesn't slip out. If you used tubing, you'll have to glue your nocks on somehow, I don't know if there's a special technique or not. The nocks must open on to the back, for reasons that should be obvious. As well as carving nocks, you can wrap a length of string around the middle of your bow to act as a handle.
Your bowstring needs to be fairly thin (2-3mm diameter), but strong, or it's likely to snap. This is an excellent time to learn to tie a bowline knot; the name says it all. The string should be slightly shorter than your bow, so that you have to bend the bow to string it. To do the actual stringing, slip both ends of the line over the end of the bow, hook one end into a nock, run the other down to the other end, bend the bow and slip it in to the other nock. When finished using the bow, always unstring it; it will prevent the bow from 'following the string' too much, which means that it'll stay straight and powerful for longer.
For arrows, I favour 8mm wooden dowel rod cut to the appropriate length for the bow. To fletch it, I first cut the nock with a tenon saw, and then take a sharp knife and make a 2-3 inch split from the base of the nock. If you don't have feathers, paper can work as a temporary fletch. You will, however, get paper cuts when shooting. If you are using feathers, cut a chevron shaped piece out of your feather and carefully glue it into the split. You want about 5mm between the end of the nock and the start of the fletch. Tie a short piece of string tightly around the gap between fletch and nock, and let dry.
This is a good time to weight the arrowhead; I don't like using pointed arrows in my back garden, so I either carefully hammer a short nail into the tip, or wrap 5 turns of 1.5mm (Your guess is as good as mine for gauge) wire around the shaft an inch back from the tip. I then wrap a few turns of electrical tape around the wire or tip, to hold it fast or prevent splitting.
You are now ready to shoot! Which you should do in one of the following manners if you do not wish to bring shame upon your house.
1. Grasp the bow firmly by the handle with your dominant hand. Nock your arrow, and hook your index and middle fingers around the string on either side of it. You should be pulling back on the string, allowing the arrow to follow. To draw the bow, step forward and lean into the bow whilst holding your non-dominant hand next to your shoulder (or as far back as you can pull it, with your dominant arm at full stretch). Release the arrow, aiming at a safe target. There should be no windows, people or animals in a 45 degree arc in front of you while you learn! Your release and aim just need to be practiced or researched until they get better.
2. As above, but grasp the bow in your non-dominant hand, and instead of stepping forward and leaning, simply pull back and release with your dominant hand. This is the more common of the two draws.
I'm not going to get into the Asian draw here, because for one thing I don't know how to use it, and for another it strikes me as less beginner-friendly than the above.
Congratulations, if you lived through the wall of text you know everything you need to become an archer with all the Arx-branded mistakes! By which I mean, caveat lector.