The first thing you need is something to record on. This could be a computer with a Digital Audio Workstation or a digital recorder. It is not recommended that you record straight to a physical medium such as a tape, because it is much harder to edit and polish your recording with this method.
Because sound waves reflect differently off of different surfaces, your room will determine the sound of your recording. For example, while the bass may sound perfect in your studio, it may be too soft in your car stereo. It's important to have some kind of acoustical treatments to combat the deceptive tendencies of your surroundings.
Microphones for recording engineers are like paint brushes for artists. When an artist wants to paint a line of a certain thickness, he has to use a certain kind of brush. Similarly, choosing the correct microphone will positively affect the sounds you record.
If you are using a computer, you will need a special interface to get the sound you are recording into the computer. These interfaces are usually high-end sound cards that connect to your computer via firewire or USB cables. A preamp amplifies the sound source so it is loud enough to record. Preamps, like microphones, will alter the sound of the signal. It's a good idea to have multiple microphones and at least one good preamp to give you a wide variety of sound.
Plug-ins are basically outboard gear for your computer. They augment your DAW (digital audio workstation); most DAWs come with some standard plugs like a basic compressor and/or equilizer.
Make sure you get yourself a nice pair of flat-response monitors and a pair of headphones. Some people have several sets of speakers to make sure they hear everything, but the same can be accomplished with one great set of speakers and headphones.