Go online or to your local music store and try to buy the best amplifier and speakers you can afford. For a home studio, you will want "near field" or close up monitors.
Make sure the amplifier has an output of twice the rated wattage of the speakers, i.e., a 300-watt amp for speakers rated at 150 watts. This allows for headroom.
Ensure that you purchase connecting cables of the proper gauge, i.e. 22 to 24 gauge for the amp input and 16 gauge and above for speakers. Check the owner's manuals for specifics.
Mount the amplifier inside a mounting rack or set it in an area where the airflow will not be blocked, as this may cause your amp to overheat.
Place speakers in positions conducive to proper audio performance, preferably in a triangular pattern away from any walls. You may also want to soundproof the front and back walls.
Connect the amplifier to the mixing board via patch cord or to your PC or Mac via firewire and speakers to the amplifier via the speaker cables.
Record some music and adjust the sound levels accordingly. Ensure the amplifier is not clipping, or distorting, and that the speakers don't rattle or distort.
Play the music back over other sound systems such as a home stereo or boom box and get a feel for how the music sounds compared to your monitors. This will help you adjust your equalization to get the best sound possible.