You need to block air travel first. If you don't seal the two spaces from each other, you will not be able to block noise. Past that there are any number of sources for products which can be used for sound blocking. How much sound (in total) are you trying to block? The level of effort and expected look in the end really depends on what you are trying to achieve. If you are going to have a band in the basement and can't hear it at all upstairs, thats vastly more difficult than just not wanting to hear someone watching TV down there. Are there speakers attached? How is the TV attached to the wall? Sound can travel through the wall as well. HVAC? Is there a straight path between the two spaces? If so, you need to change how that is laid out. Sorry, its going to get real complex, real quick. The roxul helps, but it isn't really going to make anything soundproof.

I think I've decided to just give it a shot. I plan on spending no more than a few hundred on this project. If I don't add the insulation any noise in the home will forever bother me and I will regret not trying. I accept that it may not work or that the reduction in sound may be negligible. I don't know if I will be able to do the ceiling, but I plan on doing the walls around the air ducts, bathroom walls, walls around the staircase and pipes. For pipes, any suggestion on how to insulate those? Just regular pvc. Also, are there any concerns about having this insulation around outlet boxes?

You can google soundproofing membrane, or check out this YouTube video. The video recommends using soundproofing wall board as well. You'd need to make sure to leave the minimum recommended space behind the MW. I've seen thick soundproofing insulation, but that would entail removing, and repairing, wall board behind the MW, or on the BR side.

Brad, it's not expensive to stop the sound and the noise, all you need is earplugs or noise canceling headphones that would be the best option. It gets expensive when you try to soundproof the room because the sound will travel through most solid materials and it will come out on the other side being as loud. People who do sound systems, or make in-home theaters they have a saying" to soundproof a room, you have to build another room inside it and stop the sound from one room to the other". The reason they saying that, is because the solid materials of one room cannot touch another room directly and if they do, the sound will transmit right through. So when they do soundproofing, the so-called inner room is supported by clips they called RAIC "Resilient Sound Isolation Clips" or they also have something called Whisper clip and they line the walls with a sound absorbing material. There is another way of doing this, but if you looking to soundproof something the system has to be designed, it has to be done the right way and this will cost a little. Anything else, like filling joist cavities, with Roxul, foam, etc will only muffle the sound a little, the main sound will travel through floor joists. If it was so easy everyone would be doing this. P.S Take living space above the garage, you have insulation in every bay, you have a double layer of 5/8" sheetrock, you have 3/4" plywood, you have padding and carpeting on top. When garage door opens, the sound is muffled a little, but you can still hear it in the room as you standing under garage door opener. I hope that explains a little.