"Stated somewhat simplistically, the purpose of the Golden Dawn was to provide to specially selected candidates a sytem of occult training which would enable them to acquire magical powers and eventually lead to their spiritual illumination."
The ultimate aim of the Golden Dawn system is two-fold and in two steps or stages: Knowledge & Conversation of the Holy Guardian Angel (or the Higher Self [though I technically believe the two to be different]), and, ultimately (with the previous step being the preliminary stages) Union with the Divine (although that is a more abstract philosophical pursuit, and doesn't really come into play until Adept level).
However, there are smaller, more achievable aims throughout the system. For example, one aim of the Outer Order is to give the student a thorough theoretical knowledge of magick and the occult, including ritual, divination, invocation, and so forth. This magickal training is also used as a practical means to stimulate communication and communion with the spiritual realm. The student learns techniques to heighten astral sensitivity (so that he or she can perceive things "other" to the norm), develop the intuition, clairvoyance, and other psychic faculties (primarily through divination, such as Geomancy, Astrology, and Tarot), and generally become more in tune with the spiritual and astral world. The student also learns techniques to banish the lower personality and invoke the higher, to remove astral junk, etc., and generally perfect the being through elemental and alchemical cleansing (a primary purpose of the Elemental Grades). In a sense, the purpose is to let the old self die and become spiritually reborn, with a higher understanding and a greater purpose as Inheritors of a Dying World.
The Golden Dawn is ultimately theurgical, and all the work used in the Order is designed to aid in the evolution of the soul (as well as helping others, in the Rosicrucian sense [healing the sick, etc.]). It is generally frowned upon to evoke and work with demons, pursue black magick (including curses and psychic attack), engage in sex magick, utilise spells for money or lust, and other matters revolving around the physical. One of the Order's main "slogans", from the Neophyte ceremony, is:
"Quit the Night and seek the Day!"
The meaning of this is made a little more explicit in the Theoricus ceremony:
"Quit the Material and seek the Spiritual!"