Purchase a crash cymbal. This is used to accent certain drum cues during songs with a loud, banging "crash" noise. In a kit, this will go to the left side of the drum kit for right-handed drummers and vice versa for left-handed drummers.
Purchase a floor tom. The floor tom serves the same purpose as a small bass drum in reference to everything else in a drum kit. This will go to the right side of the drum kit.
Purchase a tom-tom drum. tom-toms are two large, deep-sounding drums that go above the main snare drum in a drum kit.
Purchase a bass drum. This will be a large drum that produces low-sounding notes when struck. You can also purchase a pedal to turn your base drum into a "kick drum," which allows you to control the bass with your foot. The bass drum generally goes in front of the drummer in the kit.
Purchase a hi-hat. This is two cymbals mounted together on a stand and connected to a pedal. When engaged, the pedal triggers the cymbals. These generally go to the left of the drummer in a kit.
Purchase a snare drum. The main ingredient of any drum kit, the snare will go front and center and be the main piece of equipment the drummer uses at any given time.