Are you supposed to keep separate spindles for wool and flax. Can you use one for one material and then kosher a spindle and use it for the other  material?

Spindles may be used for wool and linen. We are prohibited from wearing Shatnez (clothing from wool and linen), but not from making clothing of Shatnez, or using the same spindle - Deuteronomy 22:11.

Nevertheless, different reasons have been suggested.

Rabbi Aaron Halevi of Barcelona wrote in his book "Sefer HaChinuch - The Book of Mitzvah Education" the reason why it is forbidden to mix wool and linen together is because it destroys the spiritual fabric of the universe. This can be explained as follows: Each and every thing on earth, except for man, has its own spiritual force that influences it. When some of these earthly items are mixed together, they cause their spiritual counterparts to become entangled. Once entangled, they cannot perform their tasks as originally designed, thusly destroying the spiritual fabric of the universe. However, after the explanation, the author tacked on "We still need a Mystic to explain this." (Sefer HaChinuch - The Book of Mitzvah Education #62)

Another explanation, from the Talmud suggests that the reason stems from the fact that when Kain and Abel brought offerings to G-d, one of them brought flax (the plant that linen is made from) and the other brought a sheep (where we get wool from). For some reason, this mixture ended up being lethal and Abel lost his life. (See Genesis 4:1-17 and the Midrash - Genesis Rabbah)

Whatever the reason, the laws of Shatnez are still applicable today, and one can find many Shatnez laboratories that can check to see if one's clothing contains Shatnez or not.

To learn more, read "Sefer HaChinuch - The Book of Mitzvah Education" (published by Feldheim), available from a Jewish bookstore or from www.feldheim.com.