One way of preventing fingerprints from showing up on your discs is by holding the disc as you place your index finger in the smaller center hole of the disc. Or you can easily hold the thin out skirts of the disc and this is a conventional way of easy holding without dropping your disk and putting fingerprints on it. Never touch the bottom section of the disc (the play side). Touching the play side of the disk could easily transfer fingerprints all over it, thus making the disc unreliable.
If you do happen to get a few fingerprints on the disc, don't worry about it. Gently run cold water through the bottom and top of the disc. Make sure that the water is running low as you do this. Then use a disc cloth or other nonabrasive cloth to dry the disk. If you use a paper towel or other course materials to dry the disk it will likely get scratched.