Hi, Chris. I would go to the site www.homasote.com and check out a product called Sound-A-Sote. I'm not sure about the best way to install it in your particular installation (drop ceiling or drywall, fiberglass insulation or not between the joists) but someone there should be able to give you some direction. Ask for sources when you ask your other questions.
For years, I used Homasote for sound deadening and know that it works. My son-in-law complained like crazy about hearing the clothes washer in the basement while he was trying to sleep in the second floor bedroom. Since we were rebuilding the stairs, I put Homasote under the stairs screwed between the risers. For architectural reasons, I put plywood over the Homasote, but drywall would have been better. No more noise.