How do I finish the area around my furnace so it looks nice and isolates noise, but is still accessible?
I take it that these are both natural gas fired and both are currently taking their combustion air from the room. It looks like the furnace could be connected to get combustion air from the outside but the PVC tubing is not connected to a tube to draw from outside. I think you should investigate getting this done.

If you box these in to form a heater closet, you must provide vents for air from the room to move into the closet. The adequacy of vents for combustion air is regulated by law and regulations. You must contact your local construction inspection authority and see what they will require for size of a heater closet, access, and venting.

It looks to me like you are possibly not leaving enough clearance on the right side of the water heater.  I have a furnace like this, in a closet, and there is still noise. However, my Carrier NG furnace is 27 years old.

Fiberglass insulation on the outgoing duct might make a difference in sound level.