You'll need at least one microphone, but there are many with different applications. A good start is a decent quality condenser for vocals and acoustic instruments, and a set of cheaper dynamic microphones for louder uses, such as drums.
Most studios use digital harddisk recording, and since its introduction in the early 1980s, it has steadily reduced in price to the extent that most other mediums, such as DAT or 2-inch tape, have now ceased to compete.
The centerpiece of your studio, your mixer should give you total control over the shaping and positioning of every element you record. Mixers generally have between 4 and 128 channels. Think about how many you really need, as unnecessary tracks can increase costs dramatically.
A compressor does just that, it compresses each sound, flattening peaks and boosting quiet sections. It is vital for mixing multiple tracks, as it helps stop each element from overpowering the others. You will use this on everything you record, so buy the best you can afford.