If cost is a big factor then you are limited to just how soundproof a room you will achieve.
The most cost effective way is as you suggested, insulation (unfaced) put inside the floor joists, make sure that when you put it in you don't leave any holes or gaps between the joists. Then over the top of the joists install drywall 1/2. If you have a drywall lift (or want to rent one) or have a few friends that want to help I'd go with 5/8 drywall, you'd be surprised the difference it will make but it is more expensive and far heavier.
