Good soundproofing is pricey and requires a lot of work with very specific materials. Fortunately, you can do a lot of good with some cheap and easy to find materials and a little ingenuity.
Use egg crate foam mattress covers. You can find them at thrift stores, and they make excellent acoustic insulators. Simply nail, glue, or staple them on to your walls.
Go a step up and use acoustic foam. It can be used in the same way as egg crate foam, but it is designed especially for the job.
Hang tapestries along the wall. Cloth tapestries are a cheap way to cover up the unseemly sight of foam covering the walls. They will also provide an additional layer of sound damping.
Use thick carpeting with a heavy foam pad. Simply put, the thicker the carpet and the heavier and denser the pad, the better the soundproofing effect will be.
Use sound baffles. Hang sheets of cloth or foam from various points in your ceiling. They do not have to go all the way from floor to ceiling to noticeably dampen sound.
Put old mattresses up against the wall. This is unseemly, takes up a lot of space, and can attract mould, but it is often the cheapest way to get the job done. It is the preferred method of garage bands everywhere.
Put your bookshelves near the wall. They should be at least an inch away from the wall, and as filled with books as possible.