Many spinners start out on a homemade CD spindle, because they're practically free; if you happen to have a suitably-sized dowel in your house, they are free.
Here's what you need: one blank CD (or maybe a software installation CD for a piece of equipment you no longer own, or a music CD you bought that you find out you hate - basically just a CD you no longer want to use for its original purpose), a dowel that fits inside the hole, and a pair of rubber bands.
Slide the CD onto the dowel, and hold it in place by wrapping the rubber bands many times around the dowel and then rolling them tightly to either side of the CD. At this point, the spindle is functional, but you will find that the rubber bands won't stay in place and you will have to keep tightening them around the CD.
Keep the rubber bands in place by putting the dowel through a roll of paper towels to hold it upright and drizzle glue down the sides of the dowel so it coats the rubber bands and forms a 'glue disk'. Let it dry, turn it over and do the other side.
You may want to use two or three CDs to get more weight on the whorl. Of course, the more weight on the spindle, the less thin you can spin, so it's a tradeoff.
And there you have it, an almost-free spindle ready to use. CD spindles are good for spinning 'knitting weight' yarn, and for plying singles into two- or three-ply yarn, but if you want to spin thin, you will need to move on eventually to a lighter weight spindle.