What's the best way to soundproof a room on budget

MASS is the way to control (reduce) TRANSMISSION of sound
from one space to another (or from outside to inside.) Adding
stiffness to the dividing surface/partition can also help, but as
you will discover, it's pretty hard to add stiffness without adding
mass at the same time -- so back to 'square one' ;~)

FUZZ is the way to control sound WITHIN a space
(reverberation, decay, reflections, etc.)

There are refinements of course, including combining the two
(when 'tuning' your listening room, for instance) but those are
the TWO BASIC principals of SOUND CONTROL. These two
principles (and the differences between them) were drummed
into my head when I studied acoustics at MIT with Bob
Newman, a principal in the famous acoustical engineering firm
of Bolt, Beranek, and Newman.

So in your case, forget the 'fuzz' (the cork and foam) at least
insofar as addressing the sound transmission problem is
concerned (you may need it to 'tune' your listening room, but
that's not what you asked about.)

Budget? The least expensive (and least invasive) way to add
mass to (YOUR side of) the wall, assuming you've permission
from the landlord, is simply to add more drywall. Either
3/4" thick, or two layers of 1/2" thick drywall. For
ease of installation, you can use 4x8' sheets for an 8' ceiling, or
4x10' sheets for a 9' ceiling. There is one very effective cost-
and-labor-reducing trick however, if you can work out the
aesthetics ;~) which is to apply the additional drywall sheets
(one layer or two) as a 4' (high) band HORIZONTALLY and
halfway between the floor and ceiling, for the length of the wall.
This puts the additional mass where it will do the most good in
terms of keeping the wall from acting like a diaphragm and
transmitting the sound to the next space. The QuietRock
product that needfreestuff mentions is excellent for new
construction, where it will be the only wall material and may
have to meet new sound transmission codes; however for your
purposes (budget and retrofitting over existing drywall) it won't
be cost-effective, compared to 'ordinary' drywall ;~)

One other thing. Do make sure there are no actual LEAKS
between the two spaces -- any place air can go, sound can go.
This problem commonly occurs in cheap construction,
especially in the form of electrical outlets (boxes) have been
installed back-to-back for the two spaces! This also applies to
floors and ceilings where there are common attic and/or crawl
spaces. It doesn't seem like it would make that big a difference,
but it does! So take off all the wall plates and caulk your brains
out!
.

If you add drywall also consider using Green Glue between the layers. It is not a glue, but rather a sound deadening compound. You can also use mass loaded vinyl.

This website has some good information on various methods and their effectiveness.

Sound Proofing Company

The biggest problem is deep bass. It is almost impossible to stop without major work.

I had Tony Manasian (of Tonian Labs) over at my apt. after I bought a pair of his speakers. His main gig is professional high end installations and consulting and he told me that bass is the hardest to control and advised me to hang some natural fibre mats or rugs behind my speaker to keep sound from transferring down to my neighbors below me.

It worked, to a point, depending on how loud I play my music. The bass is another problem since my speakers roll off around 40Hz. As for the courtyard and when I forget to close the windows, that is another matter.

Good luck,
Nonoise

If you can, stagger the studs in the wall so a stud is only connected to one panel of sheet rock. Then fill the wall with fiberglass insulation, snaking the insulation between the adjoining studs. That keeps the wall from becoming a drum, and damps out higher frequencies. We did this when building the house and it's amazing how well it works. Of course, your room will be two inches smaller :-)

Agreed - if you can do an isolated second wall that is a great solution. That option is discussed on the website I gave above. It is a great solution, although not always the easiest and less expensive.

You should also look at the resilient channel solutions, although they can be somewhat complicated to install.