Hey Brandon, you seem to be in the same position I was in when I got started a few years back!
As far as tools, I use four: 10 inch draw knife, rasp file (doesn't have to be a farrier's rasp), cabinet scraper and a chainsaw file to make nocks.
As for wood, I'd visit any local hardware store that sells boards. Home Depot sells red oak, and I've made a few nice bows from boards that I've bought there. If you can find someone who sells hardwood lumbur you can look for hickory boards or another type of oak, but it gets a bit pricier. The important thing is to find a board where the grain travels from one end to the other, or nearly so. I'd go for a 1 x 4 or something close to this if you want to take a shot at a pyramid-style bow.
I'd also get a piece of fabric for backing, and some gelatin or hide glue granules. Cotton or linen are fine, and hemp is nice as it's very strong. I'd back a board bow like this, just in case it's a little too dry or in case the grain isn't ideal. No one likes having their first bow shatter...
I'd think you should be able to get all of these tools, a couple starter boards and the fabric for less than $100.
As far as space, do you have any natural areas nearby? I used to live in a small, cramped apartment with a roommate that was less than amenable to my bow-making habits. Luckily there was a stretch of woods out behind the complex and I found a few trees with branches that held a stave pretty well while I used my draw knife and other tools. Hence most of my bow-making was done outdoors, which worked out fine. A little chilly in the winter, but hey...
Steps:
1. Lay bow out on board
2. Rough shape with draw knife and, if need be, rasp
3. Work on belly of limbs and floor tiller with draw knife (if you're comfortable doing this) or rasp
4. Once the limbs are bending a little and the bow is shaped the way you want it, scrape the bow's back, dip a long strip of the fabric in hide glue and press it against the bow's back. Add a couple extra layers if you want, although one is probably enough. All you need to do is prevent a splinter from rising.
5. Once the backing dries (2-3 weeks) finish tillering the bow. You'll need a tillering string or the bow string you'll be using.
6. Make a string, if you don't already have one, and some arrows and play safely!