for walls, the best way is to use staggered studs, so there is no direct transmission route via drywall-stud-drywall. Also add extra layer of drywall. And use insulation/batting between the studs; and seal the gaps at ceiling/floor/end of wall to reduce sound transmission.
That "best" way will cost you more and not every contractor has experience doing it. It will also cost you a small amount of space, since any wall built that way will be thicker (and cost some more $ too).
Floor - w/w carpet with good padding underneath helps a lot w/ sound transmission there. Another alternative is rubber flooring - like you see in gyms or home gyms sometimes - large interlocking panels. They come in colors so you can do a pattern (purple and yellow checkerboard!) if you want, and easily removable once you're past the playroom stage.
If you were going to do hardwood or bamboo (or other hard floor), of course you would use area rugs to help w/ sound. An alternative to hardwood nowadays is cork made for flooring - has some cushion to it and it's quieter.

Materials are important but the ratios of the rooms height to length to width is probably a whole lot more important.

http://www.realtraps.com/modecalc.htm