Yeah that would minimize it definitely.
You could also produce some panels to affix to the walls using plywood and wrapping them with batting and applying these to the walls but to really sound proof the wall you need to separate the tow surfaces as much as possible and this discussion can go many different ways and it all depends on how much expense and effort you have or want to put into the project.
Hotels and commercial spaces separate walls using hat channel or z channel. They apply this between two layers of sheet rock and call it good. Another effective sound proofing insulation is rock wool or mineral wool insulation but that is what generally goes in between the framing and is itchy as all get out. There is also special soundproofing panels that have gel applied to it but it is expensive. As far as the floor you would probably be money ahead by applying a thick carpet to the floor. You can go to greater lengths and more expense but I am assuming that you don't want to invest a life's savings in this project correct.
You could also get some OSB and some poly batting and some inexpensive fabric or if you want to spend a little more there is actually sound absorbing fabric that you can get. you could then apply the batting to the OSB and then apply the fabric on top of the OSB.
The idea is to increase the density of your surfaces while isolating layers and surfaces. Sound travels through adjacent layers so the less the layers are adjacent to one another the more effective the design will be. Again there are different ways to design this soundproofing system it is just the amount of money you want to spend on the project. Check out this link for a diagram of one sound proofing method utilizing the sound channel I spoke of earlier
You can see how they ran the sound channel perpendicular to the joist so that they could isolate the layers and this approach is critical for good sound isolation. Now if you apply the sound channel to the bottom of sheet rock you will loose some of the sound proofing properties because the solid surface will transmit the vibrations through to the other side.
I would Google soundproofing a room for more ideas and the main point I am trying to make here is what kind of expense do you want to incur and what level of alterations are you allowed and willing to make (in other words do you own the place or is this a rental unit.)
Good luck and I hope that this helps and hasn't just confused you!
Take care and be safe!