The below is only my opinions, so take that for what it's worth.
It is obviously possible to make a bow with some very primitive tools, but using some simple modern hand tools will make the project much easier, especially for a first time project.
Bare minimum required would be a good piece of wood, a coping saw, a 4 way rasp, good rope, some sort of finish, some method of applying the finish, some scrap 2"x4"s, and a good knife. Personally I don't think I'd want to sit there with a 4 way rasp and carve a bow though. I would also recommend a plane, a medium-large halfround file, a medium-large halfround rasp, a small chainsaw file, a small pulley, a hook to attach the pulley to, and sandpaper as a basic starter kit.
If starting from a log instead of dimensional lumber, you would also need an axe and wedges.
It is certainly possible to use stone age equivalents only, but this would increase the difficulty of the project substantially.
I hesitate to do this, but I think that fair warning should be made.
You mentioned price, and seems you don't know much of anything about making bows so it may be a leap, but I suspect that maybe you're interested in making your own bow to get into archery as cheaply as possible.
If this is the case, I suggest you stop right there. From what I've seen, the odds of your first few bows being decent aren't that great. It is very frustrating trying to learn how to shoot with a bow that doesn't shoot straight. I know this from experience. When you factor in the material and tool costs it will take you to finally work your skill level up to make a decent bow, and some decent arrows, it would probably be cheaper to get a used bow from somebody else.
If, on the other hand, you're after the satisfaction of conquering personal goals like I am... That's a completely different story and I would heartily recommend pursuing it. Every milestone is simply exhilarating.