I've built three different test facilities for measuring noise and so I have a bit of insight into what you're trying to do.
What you're interested in is keeping your neighbors from being annoyed by your loud noises. This means you wish to prevent transmission of noise through your walls.
The keys to preventing noise transmission through walls are:
- Mass
- Isolation
- Sealing
- Damping
and to a far lesser extent,
- reverberation
The first three are the most critical.
Mass means that all the walls must be as massive as possible so that it takes more energy to excite the walls into vibration - in your case, energy from the inside exciting the walls, which then radiate it to the exterior of the building. The gold standard here is concrete block, filled with sand - but that doesn't help ceilings, and you're not going to do it anyway. A reasonable approach is to use double or triple layers of the thickest drywall you can obtain. (The joints in each layer must be sealed before applying the next layer, but that's under "sealing".) This is something that, with forethought, can be done on ceilings as well.
Isolation means that you need to prevent direct conductive paths from the inside to the outside. If you have your little building on a slab, there's not much you can do there, but the mass of the concrete slab mitigate the fact that the edges of it do present a direct path. For walls and ceilings, you want to have two sets of studs, one for the exterior wall surfaces and one for the interior wall surfaces. They must not be tied together. Now of course if you want windows and doors, there'll be some connection at those penetrations, but a lot can be done with overlapping framing and sealing between them with a resilient caulk.
Sealing means that you must prevent direct air-to-air connections. So, for example, if you use standard electrical outlets set into the wall, you've just negated all that double thick drywall and given the sound a direct path into the wall cavity. Surface mount electrical outlets and conduits. Doors should be double - an outswinging outer door, an inswinging inner door. Seals at the doors. Ideally, windows would be an inner double-pane window mounted to the inner wall, and an outer double-pane window mounted to the outer wall, and the gap at the edge of the window sealed by the aforementioned resilient caulk.
All penetrations for ventilation are a big problem. Anything you can do to get ventilation is a direct path for noise. The design of baffled ventilation systems for these kinds of test rooms is so involved that I've just had no ventilation - you run the test, then you open the door. If you use a mini split for heating and cooling, you can seal around the penetration where the refrigerant lines go through the wall.
Damping - well, unfortunately there's little you can do with damping. Those big sheets of drywall (or exterior siding) can be treated in between layers with mastic, so they don't ring like drum heads. There's stuff specially made for this, but I can't remember the names. You can put fiberglass in the walls if you want to, but it won't do much. If you really tightly pack it, or use expanding foam, it'll help a bit. But thermal insulation is really intended to stop convection currents that set up in a hollow wall and enhance heat transfer across the wall. That's largely irrelevant to noise prevention.
Finally, we get to the thing everyone thinks of first, spends their money on, and gets the very least result - reverberation. All those little pieces of Sonex foam, egg crates, etc. - they're intended to cut reflection within a room. First of all, the little Sonex foam peaks don't do squat below a certain frequency - look it up, there's a coefficient for each frequency range, and to get effective reverberation control in bass frequencies the Sonex has to be something like 10 inches thick (and boy howdy is that ever expensive!) But let's be clear here: killing reverberation, making the room "dead", has a big effect on the way the music sounds inside the room, and makes it a much more pleasant space (up to a point) but it only reduces the energy available to be transmitted outward by a small amount. And if you want to use a surface treatment like that to reduce transmission in any meaningful way, you'll have to treat ALL the surface, not the usual four little 3 foot square patches on the walls. But, you will want to address reverberation - the best stuff is Roxul rock-wool product, which is about 4" thick and more effective than the 10" Sonex. If you're willing to do some of the other things I've mentioned and just want to kill the most annoying reverberation inside the room - not for reducing transmission because as I said it doesn't do a lot of that - then you can do the traditional things of hanging heavy draperies, putting down carpet, acoustic ceiling tiles.
